tac

Acronym

tac

Name

triacontiditeron,
pentacross (β₅),
aeropet(id),
pentachoral antiprism,
vertex figure of gee,
Gosset polytopes 2_1,1,
lattice C₅ contact polytope (span of its small roots),
equatorial cross-section of vertex-first gee

Field of sections

©

Circumradius

1/sqrt(2) = 0.707107

Inradius

1/sqrt(10) = 0.316228

Vertex figure

 ©    ©

Lace city
in approx. ASCII-art

©	o -- o3o3o4o (point) o O o -- x3o3o4o (hex) o -- o3o3o4o (point) where: o = o3o4o (point) O = x3o4o (oct)

o3o3o   x3o3o  
               
               
               
  o3o3x   o3o3o

x o3o   o x3o 
              
              
              
              
 o o3x   x o3o

Coordinates

(1, 0, 0, 0, 0)/sqrt(2) & all permutations, all changes of sign

Volume

sqrt(2)/30 = 0.047140

Surface

sqrt(5)/3 = 0.745356

Rel. Roundness

2 π² sqrt(5)/125 = 35.310570 %

General of army

(is itself convex)

Colonel of regiment

(is itself locally convex – uniform polyteral members:

by facets:	hex	pen	tho
hehad	5	16	0
tac	0	32	0
nophap	0	20	5
phap	0	12	5

)

Dual

pent

Dihedral angles

at tet between pen and pen: arccos(-3/5) = 126.869898°

Face vector

10, 40, 80, 80, 32

Confer

more general:: xPo3o...o3o4o
related segmentotera:: hexpy squete
related CRFs:: gyetac
ambification:: rat
general polytopal classes:: Wythoffian polytera orthoplex noble polytopes partial Stott expansions segmentotera fundamental lace prisms lace simplices Coxeter-Elte-Gosset polytopes Hanner polytopes
analogs:: regular orthoplex O_n Gossetic 2_n,1

External
links

Incidence matrix according to Dynkin symbol

x3o3o3o4o

. . . . . | 10 ♦  8 | 24 | 32 | 16
----------+----+----+----+----+---
x . . . . |  2 | 40 ♦  6 | 12 |  8
----------+----+----+----+----+---
x3o . . . |  3 |  3 | 80 |  4 |  4
----------+----+----+----+----+---
x3o3o . . ♦  4 |  6 |  4 | 80 |  2
----------+----+----+----+----+---
x3o3o3o . ♦  5 | 10 | 10 |  5 | 32

x3o3o3o4/3o

. . . .   . | 10 ♦  8 | 24 | 32 | 16
------------+----+----+----+----+---
x . . .   . |  2 | 40 ♦  6 | 12 |  8
------------+----+----+----+----+---
x3o . .   . |  3 |  3 | 80 |  4 |  4
------------+----+----+----+----+---
x3o3o .   . ♦  4 |  6 |  4 | 80 |  2
------------+----+----+----+----+---
x3o3o3o   . ♦  5 | 10 | 10 |  5 | 32

x3o3o3/2o4o

. . .   . . | 10 ♦  8 | 24 | 32 | 16
------------+----+----+----+----+---
x . .   . . |  2 | 40 ♦  6 | 12 |  8
------------+----+----+----+----+---
x3o .   . . |  3 |  3 | 80 |  4 |  4
------------+----+----+----+----+---
x3o3o   . . ♦  4 |  6 |  4 | 80 |  2
------------+----+----+----+----+---
x3o3o3/2o . ♦  5 | 10 | 10 |  5 | 32

x3o3o3/2o4/3o

. . .   .   . | 10 ♦  8 | 24 | 32 | 16
--------------+----+----+----+----+---
x . .   .   . |  2 | 40 ♦  6 | 12 |  8
--------------+----+----+----+----+---
x3o .   .   . |  3 |  3 | 80 |  4 |  4
--------------+----+----+----+----+---
x3o3o   .   . ♦  4 |  6 |  4 | 80 |  2
--------------+----+----+----+----+---
x3o3o3/2o   . ♦  5 | 10 | 10 |  5 | 32

x3o3/2o3o4o

. .   . . . | 10 ♦  8 | 24 | 32 | 16
------------+----+----+----+----+---
x .   . . . |  2 | 40 ♦  6 | 12 |  8
------------+----+----+----+----+---
x3o   . . . |  3 |  3 | 80 |  4 |  4
------------+----+----+----+----+---
x3o3/2o . . ♦  4 |  6 |  4 | 80 |  2
------------+----+----+----+----+---
x3o3/2o3o . ♦  5 | 10 | 10 |  5 | 32

x3o3/2o3o4/3o

. .   . .   . | 10 ♦  8 | 24 | 32 | 16
--------------+----+----+----+----+---
x .   . .   . |  2 | 40 ♦  6 | 12 |  8
--------------+----+----+----+----+---
x3o   . .   . |  3 |  3 | 80 |  4 |  4
--------------+----+----+----+----+---
x3o3/2o .   . ♦  4 |  6 |  4 | 80 |  2
--------------+----+----+----+----+---
x3o3/2o3o   . ♦  5 | 10 | 10 |  5 | 32

x3o3/2o3/2o4o

. .   .   . . | 10 ♦  8 | 24 | 32 | 16
--------------+----+----+----+----+---
x .   .   . . |  2 | 40 ♦  6 | 12 |  8
--------------+----+----+----+----+---
x3o   .   . . |  3 |  3 | 80 |  4 |  4
--------------+----+----+----+----+---
x3o3/2o   . . ♦  4 |  6 |  4 | 80 |  2
--------------+----+----+----+----+---
x3o3/2o3/2o . ♦  5 | 10 | 10 |  5 | 32

x3o3/2o3/2o4/3o

. .   .   .   . | 10 ♦  8 | 24 | 32 | 16
----------------+----+----+----+----+---
x .   .   .   . |  2 | 40 ♦  6 | 12 |  8
----------------+----+----+----+----+---
x3o   .   .   . |  3 |  3 | 80 |  4 |  4
----------------+----+----+----+----+---
x3o3/2o   .   . ♦  4 |  6 |  4 | 80 |  2
----------------+----+----+----+----+---
x3o3/2o3/2o   . ♦  5 | 10 | 10 |  5 | 32

x3/2o3o3o4o

.   . . . . | 10 ♦  8 | 24 | 32 | 16
------------+----+----+----+----+---
x   . . . . |  2 | 40 ♦  6 | 12 |  8
------------+----+----+----+----+---
x3/2o . . . |  3 |  3 | 80 |  4 |  4
------------+----+----+----+----+---
x3/2o3o . . ♦  4 |  6 |  4 | 80 |  2
------------+----+----+----+----+---
x3/2o3o3o . ♦  5 | 10 | 10 |  5 | 32

x3/2o3o3o4/3o

.   . . .   . | 10 ♦  8 | 24 | 32 | 16
--------------+----+----+----+----+---
x   . . .   . |  2 | 40 ♦  6 | 12 |  8
--------------+----+----+----+----+---
x3/2o . .   . |  3 |  3 | 80 |  4 |  4
--------------+----+----+----+----+---
x3/2o3o .   . ♦  4 |  6 |  4 | 80 |  2
--------------+----+----+----+----+---
x3/2o3o3o   . ♦  5 | 10 | 10 |  5 | 32

x3/2o3o3/2o4o

.   . .   . . | 10 ♦  8 | 24 | 32 | 16
--------------+----+----+----+----+---
x   . .   . . |  2 | 40 ♦  6 | 12 |  8
--------------+----+----+----+----+---
x3/2o .   . . |  3 |  3 | 80 |  4 |  4
--------------+----+----+----+----+---
x3/2o3o   . . ♦  4 |  6 |  4 | 80 |  2
--------------+----+----+----+----+---
x3/2o3o3/2o . ♦  5 | 10 | 10 |  5 | 32

x3/2o3o3/2o4/3o

.   . .   .   . | 10 ♦  8 | 24 | 32 | 16
----------------+----+----+----+----+---
x   . .   .   . |  2 | 40 ♦  6 | 12 |  8
----------------+----+----+----+----+---
x3/2o .   .   . |  3 |  3 | 80 |  4 |  4
----------------+----+----+----+----+---
x3/2o3o   .   . ♦  4 |  6 |  4 | 80 |  2
----------------+----+----+----+----+---
x3/2o3o3/2o   . ♦  5 | 10 | 10 |  5 | 32

x3/2o3/2o3o4o

.   .   . . . | 10 ♦  8 | 24 | 32 | 16
--------------+----+----+----+----+---
x   .   . . . |  2 | 40 ♦  6 | 12 |  8
--------------+----+----+----+----+---
x3/2o   . . . |  3 |  3 | 80 |  4 |  4
--------------+----+----+----+----+---
x3/2o3/2o . . ♦  4 |  6 |  4 | 80 |  2
--------------+----+----+----+----+---
x3/2o3/2o3o . ♦  5 | 10 | 10 |  5 | 32

x3/2o3/2o3o4/3o

.   .   . .   . | 10 ♦  8 | 24 | 32 | 16
----------------+----+----+----+----+---
x   .   . .   . |  2 | 40 ♦  6 | 12 |  8
----------------+----+----+----+----+---
x3/2o   . .   . |  3 |  3 | 80 |  4 |  4
----------------+----+----+----+----+---
x3/2o3/2o .   . ♦  4 |  6 |  4 | 80 |  2
----------------+----+----+----+----+---
x3/2o3/2o3o   . ♦  5 | 10 | 10 |  5 | 32

x3/2o3/2o3/2o4o

.   .   .   . . | 10 ♦  8 | 24 | 32 | 16
----------------+----+----+----+----+---
x   .   .   . . |  2 | 40 ♦  6 | 12 |  8
----------------+----+----+----+----+---
x3/2o   .   . . |  3 |  3 | 80 |  4 |  4
----------------+----+----+----+----+---
x3/2o3/2o   . . ♦  4 |  6 |  4 | 80 |  2
----------------+----+----+----+----+---
x3/2o3/2o3/2o . ♦  5 | 10 | 10 |  5 | 32

x3/2o3/2o3/2o4/3o

.   .   .   .   . | 10 ♦  8 | 24 | 32 | 16
------------------+----+----+----+----+---
x   .   .   .   . |  2 | 40 ♦  6 | 12 |  8
------------------+----+----+----+----+---
x3/2o   .   .   . |  3 |  3 | 80 |  4 |  4
------------------+----+----+----+----+---
x3/2o3/2o   .   . ♦  4 |  6 |  4 | 80 |  2
------------------+----+----+----+----+---
x3/2o3/2o3/2o   . ♦  5 | 10 | 10 |  5 | 32

o3o3o *b3o3x

. . .    . . | 10 ♦  8 | 24 | 32 |  8  8
-------------+----+----+----+----+------
. . .    . x |  2 | 40 ♦  6 | 12 |  4  4
-------------+----+----+----+----+------
. . .    o3x |  3 |  3 | 80 |  4 |  2  2
-------------+----+----+----+----+------
. o . *b3o3x ♦  4 |  6 |  4 | 80 |  1  1
-------------+----+----+----+----+------
o3o . *b3o3x ♦  5 | 10 | 10 |  5 | 16  *
. o3o *b3o3x ♦  5 | 10 | 10 |  5 |  * 16

   o           
  3 \          
     o---o---x 
  3 /  3  3/2  
   o

o3o3o *b3o3/2x

. . .    .   . | 10 ♦  8 | 24 | 32 |  8  8
---------------+----+----+----+----+------
. . .    .   x |  2 | 40 ♦  6 | 12 |  4  4
---------------+----+----+----+----+------
. . .    o3/2x |  3 |  3 | 80 |  4 |  2  2
---------------+----+----+----+----+------
. o . *b3o3/2x ♦  4 |  6 |  4 | 80 |  1  1
---------------+----+----+----+----+------
o3o . *b3o3/2x ♦  5 | 10 | 10 |  5 | 16  *
. o3o *b3o3/2x ♦  5 | 10 | 10 |  5 |  * 16

   o           
  3 \          
     o---o---x 
  3 / 3/2  3   
   o

o3o3o *b3/2o3x

. . .      . . | 10 ♦  8 | 24 | 32 |  8  8
---------------+----+----+----+----+------
. . .      . x |  2 | 40 ♦  6 | 12 |  4  4
---------------+----+----+----+----+------
. . .      o3x |  3 |  3 | 80 |  4 |  2  2
---------------+----+----+----+----+------
. o . *b3/2o3x ♦  4 |  6 |  4 | 80 |  1  1
---------------+----+----+----+----+------
o3o . *b3/2o3x ♦  5 | 10 | 10 |  5 | 16  *
. o3o *b3/2o3x ♦  5 | 10 | 10 |  5 |  * 16

   o           
  3 \          
     o---o---x 
  3 / 3/2 3/2  
   o

o3o3o *b3/2o3/2x

. . .      .   . | 10 ♦  8 | 24 | 32 |  8  8
-----------------+----+----+----+----+------
. . .      .   x |  2 | 40 ♦  6 | 12 |  4  4
-----------------+----+----+----+----+------
. . .      o3/2x |  3 |  3 | 80 |  4 |  2  2
-----------------+----+----+----+----+------
. o . *b3/2o3/2x ♦  4 |  6 |  4 | 80 |  1  1
-----------------+----+----+----+----+------
o3o . *b3/2o3/2x ♦  5 | 10 | 10 |  5 | 16  *
. o3o *b3/2o3/2x ♦  5 | 10 | 10 |  5 |  * 16

   o           
  3 \          
     o---o---x 
3/2 /  3   3   
   o

o3o3/2o *b3o3x

. .   .    . . | 10 ♦  8 | 24 | 32 |  8  8
---------------+----+----+----+----+------
. .   .    . x |  2 | 40 ♦  6 | 12 |  4  4
---------------+----+----+----+----+------
. .   .    o3x |  3 |  3 | 80 |  4 |  2  2
---------------+----+----+----+----+------
. o   . *b3o3x ♦  4 |  6 |  4 | 80 |  1  1
---------------+----+----+----+----+------
o3o   . *b3o3x ♦  5 | 10 | 10 |  5 | 16  *
. o3/2o *b3o3x ♦  5 | 10 | 10 |  5 |  * 16

   o           
  3 \          
     o---o---x 
3/2 /  3  3/2  
   o

o3o3/2o *b3o3/2x

. .   .    .   . | 10 ♦  8 | 24 | 32 |  8  8
-----------------+----+----+----+----+------
. .   .    .   x |  2 | 40 ♦  6 | 12 |  4  4
-----------------+----+----+----+----+------
. .   .    o3/2x |  3 |  3 | 80 |  4 |  2  2
-----------------+----+----+----+----+------
. o   . *b3o3/2x ♦  4 |  6 |  4 | 80 |  1  1
-----------------+----+----+----+----+------
o3o   . *b3o3/2x ♦  5 | 10 | 10 |  5 | 16  *
. o3/2o *b3o3/2x ♦  5 | 10 | 10 |  5 |  * 16

   o           
  3 \          
     o---o---x 
3/2 / 3/2  3   
   o

o3o3/2o *b3/2o3x

. .   .      . . | 10 ♦  8 | 24 | 32 |  8  8
-----------------+----+----+----+----+------
. .   .      . x |  2 | 40 ♦  6 | 12 |  4  4
-----------------+----+----+----+----+------
. .   .      o3x |  3 |  3 | 80 |  4 |  2  2
-----------------+----+----+----+----+------
. o   . *b3/2o3x ♦  4 |  6 |  4 | 80 |  1  1
-----------------+----+----+----+----+------
o3o   . *b3/2o3x ♦  5 | 10 | 10 |  5 | 16  *
. o3/2o *b3/2o3x ♦  5 | 10 | 10 |  5 |  * 16

   o           
  3 \          
     o---o---x 
3/2 / 3/2 3/2  
   o

o3o3/2o *b3/2o3/2x

. .   .      .   . | 10 ♦  8 | 24 | 32 |  8  8
-------------------+----+----+----+----+------
. .   .      .   x |  2 | 40 ♦  6 | 12 |  4  4
-------------------+----+----+----+----+------
. .   .      o3/2x |  3 |  3 | 80 |  4 |  2  2
-------------------+----+----+----+----+------
. o   . *b3/2o3/2x ♦  4 |  6 |  4 | 80 |  1  1
-------------------+----+----+----+----+------
o3o   . *b3/2o3/2x ♦  5 | 10 | 10 |  5 | 16  *
. o3/2o *b3/2o3/2x ♦  5 | 10 | 10 |  5 |  * 16

   o           
3/2 \          
     o---o---x 
3/2 /  3   3   
   o

o3/2o3/2o *b3o3x

.   .   .    . . | 10 ♦  8 | 24 | 32 |  8  8
-----------------+----+----+----+----+------
.   .   .    . x |  2 | 40 ♦  6 | 12 |  4  4
-----------------+----+----+----+----+------
.   .   .    o3x |  3 |  3 | 80 |  4 |  2  2
-----------------+----+----+----+----+------
.   o   . *b3o3x ♦  4 |  6 |  4 | 80 |  1  1
-----------------+----+----+----+----+------
o3/2o   . *b3o3x ♦  5 | 10 | 10 |  5 | 16  *
.   o3/2o *b3o3x ♦  5 | 10 | 10 |  5 |  * 16

   o           
3/2 \          
     o---o---x 
3/2 /  3  3/2  
   o

o3/2o3/2o *b3o3/2x

.   .   .    .   . | 10 ♦  8 | 24 | 32 |  8  8
-------------------+----+----+----+----+------
.   .   .    .   x |  2 | 40 ♦  6 | 12 |  4  4
-------------------+----+----+----+----+------
.   .   .    o3/2x |  3 |  3 | 80 |  4 |  2  2
-------------------+----+----+----+----+------
.   o   . *b3o3/2x ♦  4 |  6 |  4 | 80 |  1  1
-------------------+----+----+----+----+------
o3/2o   . *b3o3/2x ♦  5 | 10 | 10 |  5 | 16  *
.   o3/2o *b3o3/2x ♦  5 | 10 | 10 |  5 |  * 16

   o           
3/2 \          
     o---o---x 
3/2 / 3/2  3   
   o

o3/2o3/2o *b3/2o3x

.   .   .      . . | 10 ♦  8 | 24 | 32 |  8  8
-------------------+----+----+----+----+------
.   .   .      . x |  2 | 40 ♦  6 | 12 |  4  4
-------------------+----+----+----+----+------
.   .   .      o3x |  3 |  3 | 80 |  4 |  2  2
-------------------+----+----+----+----+------
.   o   . *b3/2o3x ♦  4 |  6 |  4 | 80 |  1  1
-------------------+----+----+----+----+------
o3/2o   . *b3/2o3x ♦  5 | 10 | 10 |  5 | 16  *
.   o3/2o *b3/2o3x ♦  5 | 10 | 10 |  5 |  * 16

   o           
3/2 \          
     o---o---x 
3/2 / 3/2 3/2  
   o

o3/2o3/2o *b3/2o3/2x

.   .   .      .   . | 10 ♦  8 | 24 | 32 |  8  8
---------------------+----+----+----+----+------
.   .   .      .   x |  2 | 40 ♦  6 | 12 |  4  4
---------------------+----+----+----+----+------
.   .   .      o3/2x |  3 |  3 | 80 |  4 |  2  2
---------------------+----+----+----+----+------
.   o   . *b3/2o3/2x ♦  4 |  6 |  4 | 80 |  1  1
---------------------+----+----+----+----+------
o3/2o   . *b3/2o3/2x ♦  5 | 10 | 10 |  5 | 16  *
.   o3/2o *b3/2o3/2x ♦  5 | 10 | 10 |  5 |  * 16

xo3oo3oo3ox&#x   → height = sqrt(2/5) = 0.632456
(pen || dual pen)

o.3o.3o.3o.    | 5 * ♦  4  4  0 |  6 12  6  0 | 4 12 12  4 0 | 1 4  6  4 1 0
.o3.o3.o3.o    | * 5 ♦  0  4  4 |  0  6 12  6 | 0  4 12 12 4 | 0 1  4  6 4 1
---------------+-----+----------+-------------+--------------+--------------
x. .. .. ..    | 2 0 | 10  *  * ♦  3  3  0  0 | 3  6  3  0 0 | 1 3  3  1 0 0
oo3oo3oo3oo&#x | 1 1 |  * 20  * ♦  0  3  3  0 | 0  3  6  3 0 | 0 1  3  3 1 0
.. .. .. .x    | 0 2 |  *  * 10 ♦  0  0  3  3 | 0  0  3  6 3 | 0 0  1  3 3 1
---------------+-----+----------+-------------+--------------+--------------
x.3o. .. ..    | 3 0 |  3  0  0 | 10  *  *  * | 2  2  0  0 0 | 1 2  1  0 0 0
xo .. .. ..&#x | 2 1 |  1  2  0 |  * 30  *  * | 0  2  2  0 0 | 0 1  2  1 0 0
.. .. .. ox&#x | 1 2 |  0  2  1 |  *  * 30  * | 0  0  2  2 0 | 0 0  1  2 1 0
.. .. .o3.x    | 0 3 |  0  0  3 |  *  *  * 10 | 0  0  0  2 2 | 0 0  0  1 2 1
---------------+-----+----------+-------------+--------------+--------------
x.3o.3o. ..    ♦ 4 0 |  6  0  0 |  4  0  0  0 | 5  *  *  * * | 1 1  0  0 0 0
xo3oo .. ..&#x ♦ 3 1 |  3  3  0 |  1  3  0  0 | * 20  *  * * | 0 1  1  0 0 0
xo .. .. ox&#x ♦ 2 2 |  1  4  1 |  0  2  2  0 | *  * 30  * * | 0 0  1  1 0 0
.. .. oo3ox&#x ♦ 1 3 |  0  3  3 |  0  0  3  1 | *  *  * 20 * | 0 0  0  1 1 0
.. .o3.o3.x    ♦ 0 4 |  0  0  6 |  0  0  0  4 | *  *  *  * 5 | 0 0  0  0 1 1
---------------+-----+----------+-------------+--------------+--------------
x.3o.3o.3o.    ♦ 5 0 | 10  0  0 | 10  0  0  0 | 5  0  0  0 0 | 1 *  *  * * *
xo3oo3oo ..&#x ♦ 4 1 |  6  4  0 |  4  6  0  0 | 1  4  0  0 0 | * 5  *  * * *
xo3oo .. ox&#x ♦ 3 2 |  3  6  1 |  1  6  3  0 | 0  2  3  0 0 | * * 10  * * *
xo .. oo3ox&#x ♦ 2 3 |  1  6  3 |  0  3  6  1 | 0  0  3  2 0 | * *  * 10 * *
.. oo3oo3ox&#x ♦ 1 4 |  0  4  6 |  0  0  6  4 | 0  0  0  4 1 | * *  *  * 5 *
.o3.o3.o3.x    ♦ 0 5 |  0  0 10 |  0  0  0 10 | 0  0  0  0 5 | * *  *  * * 1

or
o.3o.3o.3o.    & | 10 ♦  4  4 |  6 18 |  4 16 12 | 1  5 10
-----------------+----+-------+-------+----------+--------
x. .. .. ..    & |  2 | 20  * ♦  3  3 |  3  6  3 | 1  3  4
oo3oo3oo3oo&#x   |  2 |  * 20 ♦  0  6 |  0  6  6 | 0  2  6
-----------------+----+-------+-------+----------+--------
x.3o. .. ..    & |  3 |  3  0 | 20  * |  2  2  0 | 1  2  1
xo .. .. ..&#x & |  3 |  1  2 |  * 60 |  0  2  2 | 0  1  3
-----------------+----+-------+-------+----------+--------
x.3o.3o. ..    & ♦  4 |  6  0 |  4  0 | 10  *  * | 1  1  0
xo3oo .. ..&#x & ♦  4 |  3  3 |  1  3 |  * 40  * | 0  1  1
xo .. .. ox&#x   ♦  4 |  2  4 |  0  4 |  *  * 30 | 0  0  2
-----------------+----+-------+-------+----------+--------
x.3o.3o.3o.    & ♦  5 | 10  0 | 10  0 |  5  0  0 | 2  *  *
xo3oo3oo ..&#x & ♦  5 |  6  4 |  4  6 |  1  4  0 | * 10  *
xo3oo .. ox&#x & ♦  5 |  4  6 |  1  9 |  0  2  3 | *  * 20

oxo3ooo3ooo4ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo hex || pt)

o..3o..3o..4o..    | 1 * * ♦ 8  0 0 | 24  0  0 | 32  0  0 | 16  0
.o.3.o.3.o.4.o.    | * 8 * ♦ 1  6 1 |  6 12  6 | 12  8 12 |  8  8
..o3..o3..o4..o    | * * 1 ♦ 0  0 8 |  0  0 24 |  0  0 32 |  0 16
-------------------+-------+--------+----------+----------+------
oo.3oo.3oo.4oo.&#x | 1 1 0 | 8  * * ♦  6  0  0 | 12  0  0 |  8  0
.x. ... ... ...    | 0 2 0 | * 24 * ♦  1  4  1 |  4  4  4 |  4  4
.oo3.oo3.oo4.oo&#x | 0 1 1 | *  * 8 ♦  0  0  6 |  0  0 12 |  0  8
-------------------+-------+--------+----------+----------+------
ox. ... ... ...&#x | 1 2 0 | 2  1 0 | 24  *  * |  4  0  0 |  4  0
.x.3.o. ... ...    | 0 3 0 | 0  3 0 |  * 32  * |  1  2  1 |  2  2
.xo ... ... ...&#x | 0 2 1 | 0  1 2 |  *  * 24 |  0  0  4 |  0  4
-------------------+-------+--------+----------+----------+------
ox.3oo. ... ...&#x ♦ 1 3 0 | 3  3 0 |  3  1  0 | 32  *  * |  2  0
.x.3.o.3.o. ...    ♦ 0 4 0 | 0  6 0 |  0  4  0 |  * 16  * |  1  1
.xo3.oo ... ...&#x ♦ 0 3 1 | 0  3 3 |  0  1  3 |  *  * 32 |  0  2
-------------------+-------+--------+----------+----------+------
ox.3oo.3oo. ...&#x ♦ 1 4 0 | 4  6 0 |  6  4  0 |  4  1  0 | 16  *
.xo3.oo3.oo ...&#x ♦ 0 4 1 | 0  6 4 |  0  4  6 |  0  1  4 |  * 16

or
o..3o..3o..4o..    & | 2 * ♦  8  0 | 24  0 | 32  0 | 16
.o.3.o.3.o.4.o.      | * 8 ♦  2  6 | 12 12 | 24  8 | 16
---------------------+-----+-------+-------+-------+---
oo.3oo.3oo.4oo.&#x & | 1 1 | 16  * ♦  6  0 | 12  0 |  8
.x. ... ... ...      | 0 2 |  * 24 ♦  2  4 |  8  4 |  8
---------------------+-----+-------+-------+-------+---
ox. ... ... ...&#x & | 1 2 |  2  1 | 48  * |  4  0 |  4
.x.3.o. ... ...      | 0 3 |  0  3 |  * 32 |  2  2 |  4
---------------------+-----+-------+-------+-------+---
ox.3oo. ... ...&#x & ♦ 1 3 |  3  3 |  3  1 | 64  * |  2
.x.3.o.3.o. ...      ♦ 0 4 |  0  6 |  0  4 |  * 16 |  2
---------------------+-----+-------+-------+-------+---
ox.3oo.3oo. ...&#x & ♦ 1 4 |  4  6 |  6  4 |  4  1 | 32

qo ox3oo3oo4oo&#zx   → height = 0
(tegum sum of q-line and perp hex)
(tegum product of q-line with hex)

o. o.3o.3o.4o.    | 2 * ♦  8  0 | 24  0 | 32  0 | 16
.o .o3.o3.o4.o    | * 8 ♦  2  6 | 12 12 | 24  8 | 16
------------------+-----+-------+-------+-------+---
oo oo3oo3oo4oo&#x | 1 1 | 16  * ♦  6  0 | 12  0 |  8
.. .x .. .. ..    | 0 2 |  * 24 ♦  2  4 |  8  4 |  8
------------------+-----+-------+-------+-------+---
.. ox .. .. ..&#x | 1 2 |  2  1 | 48  * |  4  0 |  4
.. .x3.o .. ..    | 0 3 |  0  3 |  * 32 |  2  2 |  4
------------------+-----+-------+-------+-------+---
.. ox3oo .. ..&#x ♦ 1 3 |  3  3 |  3  1 | 64  * |  2
.. .x3.o3.o ..    ♦ 0 4 |  0  6 |  0  4 |  * 16 |  2
------------------+-----+-------+-------+-------+---
.. ox3oo3oo ..&#x ♦ 1 4 |  4  6 |  6  4 |  4  1 | 32

oxo3ooo3ooo *b3ooo&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || pseudo hex || pt)

o..3o..3o.. *b3o..    | 1 * * ♦ 8  0 0 | 24  0  0 | 32 0 0  0 | 8 8 0 0
.o.3.o.3.o. *b3.o.    | * 8 * ♦ 1  6 1 |  6 12  6 | 12 4 4 12 | 4 4 4 4
..o3..o3..o *b3..o    | * * 1 ♦ 0  0 8 |  0  0 24 |  0 0 0 32 | 0 0 8 8
----------------------+-------+--------+----------+-----------+--------
oo.3oo.3oo. *b3oo.&#x | 1 1 0 | 8  * * ♦  6  0  0 | 12 0 0  0 | 4 4 0 0
.x. ... ...    ...    | 0 2 0 | * 24 * ♦  1  4  1 |  4 2 2  4 | 2 2 2 2
.oo3.oo3.oo *b3.oo&#x | 0 1 1 | *  * 8 ♦  0  0  6 |  0 0 0 12 | 0 0 4 4
----------------------+-------+--------+----------+-----------+--------
ox. ... ...    ...&#x | 1 2 0 | 2  1 0 | 24  *  * |  4 0 0  0 | 2 2 0 0
.x.3.o. ...    ...    | 0 3 0 | 0  3 0 |  * 32  * |  1 1 1  1 | 1 1 1 1
.xo ... ...    ...&#x | 0 2 1 | 0  1 2 |  *  * 24 |  0 0 0  4 | 0 0 2 2
----------------------+-------+--------+----------+-----------+--------
ox.3oo. ...    ...&#x ♦ 1 3 0 | 3  3 0 |  3  1  0 | 32 * *  * | 1 1 0 0
.x.3.o.3.o.    ...    ♦ 0 4 0 | 0  6 0 |  0  4  0 |  * 8 *  * | 1 0 1 0
.x.3.o. ... *b3.o.    ♦ 0 4 0 | 0  6 0 |  0  4  0 |  * * 8  * | 0 1 0 1
.xo3.oo ...    ...&#x ♦ 0 3 1 | 0  3 3 |  0  1  3 |  * * * 32 | 0 0 1 1
----------------------+-------+--------+----------+-----------+--------
ox.3oo.3oo.    ...&#x ♦ 1 4 0 | 4  6 0 |  6  4  0 |  4 1 0  0 | 8 * * *
ox.3oo. ... *b3oo.&#x ♦ 1 4 0 | 4  6 0 |  6  4  0 |  4 0 1  0 | * 8 * *
.xo3.oo3.oo    ...&#x ♦ 0 4 1 | 0  6 4 |  0  4  6 |  0 1 0  4 | * * 8 *
.xo3.oo ... *b3.oo&#x ♦ 0 4 1 | 0  6 4 |  0  4  6 |  0 0 1  4 | * * * 8

or
o..3o..3o.. *b3o..    & | 2 * ♦  8  0 | 24  0 | 32 0 0 |  8  8
.o.3.o.3.o. *b3.o.      | * 8 ♦  2  6 | 12 12 | 24 4 4 |  8  8
------------------------+-----+-------+-------+--------+------
oo.3oo.3oo. *b3oo.&#x & | 1 1 | 16  * ♦  6  0 | 12 0 0 |  4  4
.x. ... ...    ...      | 0 2 |  * 24 ♦  2  4 |  8 2 2 |  4  4
------------------------+-----+-------+-------+--------+------
ox. ... ...    ...&#x & | 1 2 |  2  1 | 48  * |  4 0 0 |  2  2
.x.3.o. ...    ...      | 0 3 |  0  3 |  * 32 |  2 1 1 |  2  2
------------------------+-----+-------+-------+--------+------
ox.3oo. ...    ...&#x & ♦ 1 3 |  3  3 |  3  1 | 64 * * |  1  1
.x.3.o.3.o.    ...      ♦ 0 4 |  0  6 |  0  4 |  * 8 * |  2  0
.x.3.o. ... *b3.o.      ♦ 0 4 |  0  6 |  0  4 |  * * 8 |  0  2
------------------------+-----+-------+-------+--------+------
ox.3oo.3oo.    ...&#x & ♦ 1 4 |  4  6 |  6  4 |  4 1 0 | 16  *
ox.3oo. ... *b3oo.&#x & ♦ 1 4 |  4  6 |  6  4 |  4 0 1 |  * 16

qo ox3oo3oo *b3oo&#zx   → height = 0
(tegum sum of q-line and perp hex)
(tegum product of q-line with hex)

o. o.3o.3o. *b3o.    | 2 * ♦  8  0 | 24  0 | 32 0 0 |  8  8
.o .o3.o3.o *b3.o    | * 8 ♦  2  6 | 12 12 | 24 4 4 |  8  8
---------------------+-----+-------+-------+--------+------
oo oo3oo3oo *b3oo&#x | 1 1 | 16  * ♦  6  0 | 12 0 0 |  4  4
.. .x .. ..    ..    | 0 2 |  * 24 ♦  2  4 |  8 2 2 |  4  4
---------------------+-----+-------+-------+--------+------
.. ox .. ..    ..&#x | 1 2 |  2  1 | 48  * |  4 0 0 |  2  2
.. .x3.o ..    ..    | 0 3 |  0  3 |  * 32 |  2 1 1 |  2  2
---------------------+-----+-------+-------+--------+------
.. ox3oo ..    ..&#x ♦ 1 3 |  3  3 |  3  1 | 64 * * |  1  1
.. .x3.o3.o    ..    ♦ 0 4 |  0  6 |  0  4 |  * 8 * |  2  0
.. .x3.o .. *b3.o    ♦ 0 4 |  0  6 |  0  4 |  * * 8 |  0  2
---------------------+-----+-------+-------+--------+------
.. ox3oo3oo    ..&#x ♦ 1 4 |  4  6 |  6  4 |  4 1 0 | 16  *
.. ox3oo .. *b3oo&#x ♦ 1 4 |  4  6 |  6  4 |  4 0 1 |  * 16

xox oxo3ooo4ooo&#xt   → both heights = 1/2
(line || perp pseudo oct || line)

o.. o..3o..4o..    | 2 * * ♦ 1  6 1  0  0 0 | 6 12  6 0 0  0 | 12  8 12  0  0 | 8  8 0
.o. .o.3.o.4.o.    | * 6 * ♦ 0  2 0  4  2 0 | 1  8  2 4 1  8 |  4  8  8  4  8 | 4  8 4
..o ..o3..o4..o    | * * 2 ♦ 0  0 1  0  6 1 | 0  0  6 0 6 12 |  0  0 12 12  8 | 0  8 8
-------------------+-------+----------------+----------------+----------------+-------
x.. ... ... ...    | 2 0 0 | 1  * *  *  * * ♦ 6  0  0 0 0  0 | 12  0  0  0  0 | 8  0 0
oo. oo.3oo.4oo.&#x | 1 1 0 | * 12 *  *  * * ♦ 1  4  1 0 0  0 |  4  4  4  0  0 | 4  4 0
o.o o.o3o.o4o.o&#x | 1 0 1 | *  * 2  *  * * ♦ 0  0  6 0 0  0 |  0  0 12  0  0 | 0  8 0
... .x. ... ...    | 0 2 0 | *  * * 12  * * ♦ 0  2  0 2 0  2 |  1  4  2  1  4 | 2  4 2
.oo .oo3.oo4.oo&#x | 0 1 1 | *  * *  * 12 * ♦ 0  0  1 0 1  4 |  0  0  4  4  4 | 0  4 4
..x ... ... ...    | 0 0 2 | *  * *  *  * 1 ♦ 0  0  0 0 6  0 |  0  0  0 12  0 | 0  0 8
-------------------+-------+----------------+----------------+----------------+-------
xo. ... ... ...&#x | 2 1 0 | 1  2 0  0  0 0 | 6  *  * * *  * |  4  0  0  0  0 | 4  0 0
... ox. ... ...&#x | 1 2 0 | 0  2 0  1  0 0 | * 24  * * *  * |  1  2  1  0  0 | 2  2 0
ooo ooo3ooo4ooo&#x | 1 1 1 | 0  1 1  0  1 0 | *  * 12 * *  * |  0  0  4  0  0 | 0  4 0
... .x.3.o. ...    | 0 3 0 | 0  0 0  3  0 0 | *  *  * 8 *  * |  0  2  0  0  2 | 1  2 1
.ox ... ... ...&#x | 0 1 2 | 0  0 0  0  2 1 | *  *  * * 6  * |  0  0  0  4  0 | 0  0 4
... .xo ... ...&#x | 0 2 1 | 0  0 0  1  2 0 | *  *  * * * 24 |  0  0  1  1  2 | 0  2 2
-------------------+-------+----------------+----------------+----------------+-------
xo. ox. ... ...&#x ♦ 2 2 0 | 1  4 0  1  0 0 | 2  2  0 0 0  0 | 12  *  *  *  * | 2  0 0
... ox.3oo. ...&#x ♦ 1 3 0 | 0  3 0  3  0 0 | 0  3  0 1 0  0 |  * 16  *  *  * | 1  1 0
... oxo ... ...&#x ♦ 1 2 1 | 0  2 1  1  2 0 | 0  1  2 0 0  1 |  *  * 24  *  * | 0  2 0
.ox .xo ... ...&#x ♦ 0 2 2 | 0  0 0  1  4 1 | 0  0  0 0 2  2 |  *  *  * 12  * | 0  0 2
... .xo3.oo ...&#x ♦ 0 3 1 | 0  0 0  3  3 0 | 0  0  0 1 0  3 |  *  *  *  * 16 | 0  1 1
-------------------+-------+----------------+----------------+----------------+-------
xo. ox.3oo. ...&#x ♦ 2 3 0 | 1  6 0  3  0 0 | 3  6  0 1 0  0 |  3  2  0  0  0 | 8  * *
... oxo3ooo ...&#x ♦ 1 3 1 | 0  3 1  3  3 0 | 0  3  3 1 0  3 |  0  1  3  0  1 | * 16 *
.ox .xo3.oo ...&#x ♦ 0 3 2 | 0  0 0  3  6 1 | 0  0  0 1 3  6 |  0  0  0  3  2 | *  * 8

or
o.. o..3o..4o..     & | 4 * ♦ 1  6 1  0 |  6 12  6 0 | 12  8 12 |  8  8
.o. .o.3.o.4.o.       | * 6 ♦ 0  4 0  4 |  2 16  2 4 |  8 16  8 |  8  8
----------------------+-----+-----------+------------+----------+------
x.. ... ... ...     & | 2 0 | 2  * *  * ♦  6  0  0 0 | 12  0  0 |  8  0
oo. oo.3oo.4oo.&#x  & | 1 1 | * 24 *  * ♦  1  4  1 0 |  4  4  4 |  4  4
o.o o.o3o.o4o.o&#x    | 2 0 | *  * 2  * ♦  0  0  6 0 |  0  0 12 |  0  8
... .x. ... ...       | 0 2 | *  * * 12 ♦  0  4  0 2 |  2  8  2 |  4  4
----------------------+-----+-----------+------------+----------+------
xo. ... ... ...&#x  & | 2 1 | 1  2 0  0 | 12  *  * * |  4  0  0 |  4  0
... ox. ... ...&#x  & | 1 2 | 0  2 0  1 |  * 48  * * |  1  2  1 |  2  2
ooo ooo3ooo4ooo&#x    | 2 1 | 0  2 1  0 |  *  * 12 * |  0  0  4 |  0  4
... .x.3.o. ...       | 0 3 | 0  0 0  3 |  *  *  * 8 |  0  4  0 |  2  2
----------------------+-----+-----------+------------+----------+------
xo. ox. ... ...&#x  & ♦ 2 2 | 1  4 0  1 |  2  2  0 0 | 24  *  * |  2  0
... ox.3oo. ...&#x  & ♦ 1 3 | 0  3 0  3 |  0  3  0 1 |  * 32  * |  1  1
... oxo ... ...&#x    ♦ 2 2 | 0  4 1  1 |  0  2  2 0 |  *  * 24 |  0  2
----------------------+-----+-----------+------------+----------+------
xo. ox.3oo. ...&#x  & ♦ 2 3 | 1  6 0  3 |  3  6  0 1 |  3  2  0 | 16  *
... oxo3ooo ...&#x    ♦ 2 3 | 0  6 1  3 |  0  6  3 1 |  0  2  3 |  * 16

xox ooo3oxo3ooo&#xt   → both heights = 1/2
(line || perp pseudo oct || line)

o.. o..3o..3o..    | 2 * * ♦ 1  6 1  0  0 0 | 6 12  6 0 0 0  0 | 12 4 4 12  0 0 0 | 4 4 4 4 0 0
.o. .o.3.o.3.o.    | * 6 * ♦ 0  2 0  4  2 0 | 1  8  2 2 2 1  8 |  4 4 4  8  4 4 4 | 2 2 4 4 2 2
..o ..o3..o3..o    | * * 2 ♦ 0  0 1  0  6 1 | 0  0  6 0 0 6 12 |  0 0 0 12 12 4 4 | 0 0 4 4 4 4
-------------------+-------+----------------+------------------+------------------+------------
x.. ... ... ...    | 2 0 0 | 1  * *  *  * * ♦ 6  0  0 0 0 0  0 | 12 0 0  0  0 0 0 | 4 4 0 0 0 0
oo. oo.3oo.3oo.&#x | 1 1 0 | * 12 *  *  * * ♦ 1  4  1 0 0 0  0 |  4 2 2  4  0 0 0 | 2 2 2 2 0 0
o.o o.o3o.o3o.o&#x | 1 0 1 | *  * 2  *  * * ♦ 0  0  6 0 0 0  0 |  0 0 0 12  0 0 0 | 0 0 4 4 0 0
... ... .x. ...    | 0 2 0 | *  * * 12  * * ♦ 0  2  0 1 1 0  2 |  1 2 2  2  1 2 2 | 1 1 2 2 1 1
.oo .oo3.oo3.oo&#x | 0 1 1 | *  * *  * 12 * ♦ 0  0  1 0 0 1  4 |  0 0 0  4  4 2 2 | 0 0 2 2 2 2
..x ... ... ...    | 0 0 2 | *  * *  *  * 1 ♦ 0  0  0 0 0 6  0 |  0 0 0  0 12 0 0 | 0 0 0 0 4 4
-------------------+-------+----------------+------------------+------------------+------------
xo. ... ... ...&#x | 2 1 0 | 1  2 0  0  0 0 | 6  *  * * * *  * |  4 0 0  0  0 0 0 | 2 2 0 0 0 0
... ... ox. ...&#x | 1 2 0 | 0  2 0  1  0 0 | * 24  * * * *  * |  1 1 1  1  0 0 0 | 1 1 1 1 0 0
ooo ooo3ooo3ooo&#x | 1 1 1 | 0  1 1  0  1 0 | *  * 12 * * *  * |  0 0 0  4  0 0 0 | 0 0 2 2 0 0
... .o.3.x. ...    | 0 3 0 | 0  0 0  3  0 0 | *  *  * 4 * *  * |  0 2 0  0  0 2 0 | 1 0 2 0 1 0
... ... .x.3.o.    | 0 3 0 | 0  0 0  3  0 0 | *  *  * * 4 *  * |  0 0 2  0  0 0 2 | 0 1 0 2 0 1
.ox ... ... ...&#x | 0 1 2 | 0  0 0  0  2 1 | *  *  * * * 6  * |  0 0 0  0  4 0 0 | 0 0 0 0 2 2
... ... .xo ...&#x | 0 2 1 | 0  0 0  1  2 0 | *  *  * * * * 24 |  0 0 0  1  1 1 1 | 0 0 1 1 1 1
-------------------+-------+----------------+------------------+------------------+------------
xo. ... ox. ...&#x ♦ 2 2 0 | 1  4 0  1  0 0 | 2  2  0 0 0 0  0 | 12 * *  *  * * * | 1 1 0 0 0 0
... oo.3ox. ...&#x ♦ 1 3 0 | 0  3 0  3  0 0 | 0  3  0 1 0 0  0 |  * 8 *  *  * * * | 1 0 1 0 0 0
... ... ox.3oo.&#x ♦ 1 3 0 | 0  3 0  3  0 0 | 0  3  0 0 1 0  0 |  * * 8  *  * * * | 0 1 0 1 0 0
... ... oxo ...&#x ♦ 1 2 1 | 0  2 1  1  2 0 | 0  1  2 0 0 0  1 |  * * * 24  * * * | 0 0 1 1 0 0
.ox ... .xo ...&#x ♦ 0 2 2 | 0  0 0  1  4 1 | 0  0  0 0 0 2  2 |  * * *  * 12 * * | 0 0 0 0 1 1
... .oo3.xo ...&#x ♦ 0 3 1 | 0  0 0  3  3 0 | 0  0  0 1 0 0  3 |  * * *  *  * 8 * | 0 0 1 0 1 0
... ... .xo3.oo&#x ♦ 0 3 1 | 0  0 0  3  3 0 | 0  0  0 0 1 0  3 |  * * *  *  * * 8 | 0 0 0 1 0 1
-------------------+-------+----------------+------------------+------------------+------------
xo. oo.3ox. ...&#x ♦ 2 3 0 | 1  6 0  3  0 0 | 3  6  0 1 0 0  0 |  3 2 0  0  0 0 0 | 4 * * * * *
xo. ... ox.3oo.&#x ♦ 2 3 0 | 1  6 0  3  0 0 | 3  6  0 0 1 0  0 |  3 0 2  0  0 0 0 | * 4 * * * *
... ooo3oxo ...&#x ♦ 1 3 1 | 0  3 1  3  3 0 | 0  3  3 1 0 0  3 |  0 1 0  3  0 1 0 | * * 8 * * *
... ... oxo3ooo&#x ♦ 1 3 1 | 0  3 1  3  3 0 | 0  3  3 0 1 0  3 |  0 0 1  3  0 0 1 | * * * 8 * *
.ox .oo3.xo ...&#x ♦ 0 3 2 | 0  0 0  3  6 1 | 0  0  0 1 0 3  6 |  0 0 0  0  3 2 0 | * * * * 4 *
.ox ... .xo3.oo&#x ♦ 0 3 2 | 0  0 0  3  6 1 | 0  0  0 0 1 3  6 |  0 0 0  0  3 0 2 | * * * * * 4

or
o.. o..3o..3o..     & | 4 * ♦ 1  6 1  0 |  6 12  6 0 0 | 12  4  4 12 | 4 4 4 4
.o. .o.3.o.3.o.       | * 6 ♦ 0  4 0  4 |  2 16  2 2 2 |  8  8  8  8 | 4 4 4 4
----------------------+-----+-----------+--------------+-------------+--------
x.. ... ... ...     & | 2 0 | 2  * *  * ♦  6  0  0 0 0 | 12  0  0  0 | 4 4 0 0
oo. oo.3oo.3oo.&#x  & | 1 1 | * 24 *  * ♦  1  4  1 0 0 |  4  2  2  4 | 2 2 2 2
o.o o.o3o.o3o.o&#x    | 2 0 | *  * 2  * ♦  0  0  6 0 0 |  0  0  0 12 | 0 0 4 4
... ... .x. ...       | 0 2 | *  * * 12 ♦  0  4  0 1 1 |  2  4  4  2 | 2 2 2 2
----------------------+-----+-----------+--------------+-------------+--------
xo. ... ... ...&#x  & | 2 1 | 1  2 0  0 | 12  *  * * * |  4  0  0  0 | 2 2 0 0
... ... ox. ...&#x  & | 1 2 | 0  2 0  1 |  * 48  * * * |  1  1  1  1 | 1 1 1 1
ooo ooo3ooo3ooo&#x    | 2 1 | 0  2 1  0 |  *  * 12 * * |  0  0  0  4 | 0 0 2 2
... .o.3.x. ...       | 0 3 | 0  0 0  3 |  *  *  * 4 * |  0  4  0  0 | 2 0 2 0
... ... .x.3.o.       | 0 3 | 0  0 0  3 |  *  *  * * 4 |  0  0  4  0 | 0 2 0 2
----------------------+-----+-----------+--------------+-------------+--------
xo. ... ox. ...&#x  & ♦ 2 2 | 1  4 0  1 |  2  2  0 0 0 | 24  *  *  * | 1 1 0 0
... oo.3ox. ...&#x  & ♦ 1 3 | 0  3 0  3 |  0  3  0 1 0 |  * 16  *  * | 1 0 1 0
... ... ox.3oo.&#x  & ♦ 1 3 | 0  3 0  3 |  0  3  0 0 1 |  *  * 16  * | 0 1 0 1
... ... oxo ...&#x    ♦ 2 2 | 0  4 1  1 |  0  2  2 0 0 |  *  *  * 24 | 0 0 1 1
----------------------+-----+-----------+--------------+-------------+--------
xo. oo.3ox. ...&#x  & ♦ 2 3 | 1  6 0  3 |  3  6  0 1 0 |  3  2  0  0 | 8 * * *
xo. ... ox.3oo.&#x  & ♦ 2 3 | 1  6 0  3 |  3  6  0 0 1 |  3  0  2  0 | * 8 * *
... ooo3oxo ...&#x    ♦ 2 3 | 0  6 1  3 |  0  6  3 1 0 |  0  2  0  3 | * * 8 *
... ... oxo3ooo&#x    ♦ 2 3 | 0  6 1  3 |  0  6  3 0 1 |  0  0  2  3 | * * * 8

xoo3oox oxo4ooo&#xt   → both heights = 1/sqrt(6) = 0.408248
({3} || perp pseudo {4} || dual {3})

o..3o.. o..4o..    | 3 * * ♦ 2  4 2 0  0 0 | 1  8  4 2 1  8  0  0 0 | 4  8  8  4  8 0  0 | 4  8  4 0
.o.3.o. .o.4.o.    | * 4 * ♦ 0  3 0 2  3 0 | 0  3  6 0 0  6  3  6 0 | 1  6  3  3 12 1  6 | 2  6  6 2
..o3..o ..o4..o    | * * 3 ♦ 0  0 2 0  4 2 | 0  0  0 1 2  8  8  4 1 | 0  0  4  8  8 4  8 | 0  4  8 4
-------------------+-------+---------------+------------------------+--------------------+----------
x.. ... ... ...    | 2 0 0 | 3  * * *  * * ♦ 1  4  0 1 0  0  0  0 0 | 4  4  4  0  0 0  0 | 4  4  0 0
oo.3oo. oo.4oo.&#x | 1 1 0 | * 12 * *  * * ♦ 0  2  2 0 0  2  0  0 0 | 1  4  2  1  4 0  0 | 2  4  2 0
o.o3o.o o.o4o.o&#x | 1 0 1 | *  * 6 *  * * ♦ 0  0  0 1 1  4  0  0 0 | 0  0  4  4  4 0  0 | 0  4  4 0
... ... .x. ...    | 0 2 0 | *  * * 4  * * ♦ 0  0  3 0 0  0  0  3 0 | 0  3  0  0  6 0  3 | 1  3  3 1
.oo3.oo .oo4.oo&#x | 0 1 1 | *  * * * 12 * ♦ 0  0  0 0 0  2  2  2 0 | 0  0  1  2  4 1  4 | 0  2  4 2
... ..x ... ...    | 0 0 2 | *  * * *  * 3 ♦ 0  0  0 0 1  0  4  0 1 | 0  0  0  4  0 4  4 | 0  0  4 4
-------------------+-------+---------------+------------------------+--------------------+----------
x..3o.. ... ...    | 3 0 0 | 3  0 0 0  0 0 | 1  *  * * *  *  *  * * | 4  0  0  0  0 0  0 | 4  0  0 0
xo. ... ... ...&#x | 2 1 0 | 1  2 0 0  0 0 | * 12  * * *  *  *  * * | 1  2  1  0  0 0  0 | 2  2  0 0
... ... ox. ...&#x | 1 2 0 | 0  2 0 1  0 0 | *  * 12 * *  *  *  * * | 0  2  0  0  2 0  0 | 1  2  1 0
x.o ... ... ...&#x | 2 0 1 | 1  0 2 0  0 0 | *  *  * 3 *  *  *  * * | 0  0  4  0  0 0  0 | 0  4  0 0
... o.x ... ...&#x | 1 0 2 | 0  0 2 0  0 1 | *  *  * * 3  *  *  * * | 0  0  0  4  0 0  0 | 0  0  4 0
ooo3ooo ooo4ooo&#x | 1 1 1 | 0  1 1 0  1 0 | *  *  * * * 24  *  * * | 0  0  1  1  2 0  0 | 0  2  2 0
... .ox ... ...&#x | 0 1 2 | 0  0 0 0  2 1 | *  *  * * *  * 12  * * | 0  0  0  1  0 1  2 | 0  0  2 2
... ... .xo ...&#x | 0 2 1 | 0  0 0 1  2 0 | *  *  * * *  *  * 12 * | 0  0  0  0  2 0  2 | 0  1  2 1
..o3..x ... ...    | 0 0 3 | 0  0 0 0  0 3 | *  *  * * *  *  *  * 1 | 0  0  0  0  0 4  0 | 0  0  0 4
-------------------+-------+---------------+------------------------+--------------------+----------
xo.3oo. ... ...&#x ♦ 3 1 0 | 3  3 0 0  0 0 | 1  3  0 0 0  0  0  0 0 | 4  *  *  *  * *  * | 2  0  0 0
xo. ... ox. ...&#x ♦ 2 2 0 | 1  4 0 1  0 0 | 0  2  2 0 0  0  0  0 0 | * 12  *  *  * *  * | 1  1  0 0
xoo ... ... ...&#x ♦ 2 1 1 | 1  2 2 0  1 0 | 0  1  0 1 0  2  0  0 0 | *  * 12  *  * *  * | 0  2  0 0
... oox ... ...&#x ♦ 1 1 2 | 0  1 2 0  2 1 | 0  0  0 0 1  2  1  0 0 | *  *  * 12  * *  * | 0  0  2 0
... ... oxo ...&#x ♦ 1 2 1 | 0  2 1 1  2 0 | 0  0  1 0 0  2  0  1 0 | *  *  *  * 24 *  * | 0  1  1 0
.oo3.ox ... ...&#x ♦ 0 1 3 | 0  0 0 0  3 3 | 0  0  0 0 0  0  3  0 1 | *  *  *  *  * 4  * | 0  0  0 2
... .ox .xo ...&#x ♦ 0 2 2 | 0  0 0 1  4 1 | 0  0  0 0 0  0  2  2 0 | *  *  *  *  * * 12 | 0  0  1 1
-------------------+-------+---------------+------------------------+--------------------+----------
xo.3oo. ox. ...&#x ♦ 3 2 0 | 3  6 0 1  0 0 | 1  6  3 0 0  0  0  0 0 | 2  3  0  0  0 0  0 | 4  *  * *
xoo ... oxo ...&#x ♦ 2 2 1 | 1  4 2 1  2 0 | 0  2  2 1 0  4  0  1 0 | 0  1  2  0  2 0  0 | * 12  * *
... oox oxo ...&#x ♦ 1 2 2 | 0  2 2 1  4 1 | 0  0  1 0 1  4  2  2 0 | 0  0  0  2  2 0  1 | *  * 12 *
.oo3.ox .xo ...&#x ♦ 0 2 3 | 0  0 0 1  6 3 | 0  0  0 0 0  0  6  3 1 | 0  0  0  0  0 2  3 | *  *  * 4

or
o..3o.. o..4o..    & | 6 * ♦ 2  4 2 0 | 1  8  4 3  8 | 4  8 12  8 | 4 12
.o.3.o. .o.4.o.      | * 4 ♦ 0  6 0 2 | 0  6 12 0  6 | 2 12  6 12 | 4 12
---------------------+-----+----------+--------------+------------+-----
x.. ... ... ...    & | 2 0 | 6  * * * ♦ 1  4  0 1  0 | 4  4  4  0 | 4  4
oo.3oo. oo.4oo.&#x & | 1 1 | * 24 * * ♦ 0  2  2 0  2 | 1  4  3  4 | 2  6
o.o3o.o o.o4o.o&#x   | 2 0 | *  * 6 * ♦ 0  0  0 2  4 | 0  0  8  4 | 0  8
... ... .x. ...      | 0 2 | *  * * 4 ♦ 0  0  6 0  0 | 0  6  0  6 | 2  6
---------------------+-----+----------+--------------+------------+-----
x..3o.. ... ...    & | 3 0 | 3  0 0 0 | 2  *  * *  * | 4  0  0  0 | 4  0
xo. ... ... ...&#x & | 2 1 | 1  2 0 0 | * 24  * *  * | 1  2  1  0 | 2  2
... ... ox. ...&#x & | 1 2 | 0  2 0 1 | *  * 24 *  * | 0  2  0  2 | 1  3
x.o ... ... ...&#x & | 3 0 | 1  0 2 0 | *  *  * 6  * | 0  0  4  0 | 0  4
ooo3ooo ooo4ooo&#x   | 2 1 | 0  2 1 0 | *  *  * * 24 | 0  0  2  2 | 0  4
---------------------+-----+----------+--------------+------------+-----
xo.3oo. ... ...&#x & ♦ 3 1 | 3  3 0 0 | 1  3  0 0  0 | 8  *  *  * | 2  0
xo. ... ox. ...&#x & ♦ 2 2 | 1  4 0 1 | 0  2  2 0  0 | * 24  *  * | 1  1
xoo ... ... ...&#x & ♦ 3 1 | 1  3 2 0 | 0  1  0 1  2 | *  * 24  * | 0  2
... ... oxo ...&#x   ♦ 2 2 | 0  4 1 1 | 0  0  2 0  2 | *  *  * 24 | 0  2
---------------------+-----+----------+--------------+------------+-----
xo.3oo. ox. ...&#x & ♦ 3 2 | 3  6 0 1 | 1  6  3 0  0 | 2  3  0  0 | 8  *
xoo ... oxo ...&#x & ♦ 3 2 | 1  6 2 1 | 0  2  3 1  4 | 0  1  2  2 | * 24

oxo oxo xoo3oox&#xt   → both heights = 1/sqrt(6) = 0.408248
({3} || perp pseudo {4} || dual {3})

o.. o.. o..3o..    | 3 * * ♦ 2  4 2 0 0  0 0 | 1 2 2  8 2 1  8 0 0  0 0 | 4 4 4  4  4  8  4 0 0 0 | 2 2 4 2 4 2 0 0
.o. .o. .o.3.o.    | * 4 * ♦ 0  3 0 1 1  3 0 | 0 3 3  3 0 0  6 3 3  3 0 | 3 3 1  6  6  3  3 3 3 1 | 1 1 3 3 3 3 1 1
..o ..o ..o3..o    | * * 3 ♦ 0  0 2 0 0  4 2 | 0 0 0  0 1 2  8 2 2  8 1 | 0 0 0  4  4  4  8 4 4 4 | 0 0 2 4 2 4 2 2
-------------------+-------+-----------------+--------------------------+-------------------------+----------------
... ... x.. ...    | 2 0 0 | 3  * * * *  * * ♦ 1 0 0  4 1 0  0 0 0  0 0 | 2 2 4  0  0  4  0 0 0 0 | 2 2 2 0 2 0 0 0
oo. oo. oo.3oo.&#x | 1 1 0 | * 12 * * *  * * ♦ 0 1 1  2 0 0  2 0 0  0 0 | 2 2 1  2  2  2  1 0 0 0 | 1 1 2 1 2 1 0 0
o.o o.o o.o3o.o&#x | 1 0 1 | *  * 6 * *  * * ♦ 0 0 0  0 1 1  4 0 0  0 0 | 0 0 0  2  2  4  4 0 0 0 | 0 0 2 2 2 2 0 0
.x. ... ... ...    | 0 2 0 | *  * * 2 *  * * ♦ 0 3 0  0 0 0  0 3 0  0 0 | 3 0 0  6  0  0  0 3 0 0 | 1 0 3 3 0 0 1 0
... .x. ... ...    | 0 2 0 | *  * * * 2  * * ♦ 0 0 3  0 0 0  0 0 3  0 0 | 0 3 0  0  6  0  0 0 3 0 | 0 1 0 0 3 3 0 1
.oo .oo .oo3.oo&#x | 0 1 1 | *  * * * * 12 * ♦ 0 0 0  0 0 0  2 1 1  2 0 | 0 0 0  2  2  1  2 2 2 1 | 0 0 1 2 1 2 1 1
... ... ... ..x    | 0 0 2 | *  * * * *  * 3 ♦ 0 0 0  0 0 1  0 0 0  4 1 | 0 0 0  0  0  0  4 2 2 4 | 0 0 0 2 0 2 2 2
-------------------+-------+-----------------+--------------------------+-------------------------+----------------
... ... x..3o..    | 3 0 0 | 3  0 0 0 0  0 0 | 1 * *  * * *  * * *  * * | 0 0 4  0  0  0  0 0 0 0 | 2 2 0 0 0 0 0 0
ox. ... ... ...&#x | 1 2 0 | 0  2 0 1 0  0 0 | * 6 *  * * *  * * *  * * | 2 0 0  2  0  0  0 0 0 0 | 1 0 2 1 0 0 0 0
... ox. ... ...&#x | 1 2 0 | 0  2 0 0 1  0 0 | * * 6  * * *  * * *  * * | 0 2 0  0  2  0  0 0 0 0 | 0 1 0 0 2 1 0 0
... ... xo. ...&#x | 2 1 0 | 1  2 0 0 0  0 0 | * * * 12 * *  * * *  * * | 1 1 1  0  0  1  0 0 0 0 | 1 1 1 0 1 0 0 0
... ... x.o ...&#x | 2 0 1 | 1  0 2 0 0  0 0 | * * *  * 3 *  * * *  * * | 0 0 0  0  0  4  0 0 0 0 | 0 0 2 0 2 0 0 0
... ... ... o.x&#x | 1 0 2 | 0  0 2 0 0  0 1 | * * *  * * 3  * * *  * * | 0 0 0  0  0  0  4 0 0 0 | 0 0 0 2 0 2 0 0
ooo ooo ooo3ooo&#x | 1 1 1 | 0  1 1 0 0  1 0 | * * *  * * * 24 * *  * * | 0 0 0  1  1  1  1 0 0 0 | 0 0 1 1 1 1 0 0
.xo ... ... ...&#x | 0 2 1 | 0  0 0 1 0  2 0 | * * *  * * *  * 6 *  * * | 0 0 0  2  0  0  0 2 0 0 | 0 0 1 2 0 0 1 0
... .xo ... ...&#x | 0 2 1 | 0  0 0 0 1  2 0 | * * *  * * *  * * 6  * * | 0 0 0  0  2  0  0 0 2 0 | 0 0 0 0 1 2 0 1
... ... ... .ox&#x | 0 1 2 | 0  0 0 0 0  2 1 | * * *  * * *  * * * 12 * | 0 0 0  0  0  0  1 1 1 1 | 0 0 0 1 0 1 1 1
... ... ..o3..x    | 0 0 3 | 0  0 0 0 0  0 3 | * * *  * * *  * * *  * 1 | 0 0 0  0  0  0  0 0 0 4 | 0 0 0 0 0 0 2 2
-------------------+-------+-----------------+--------------------------+-------------------------+----------------
ox. ... xo. ...&#x ♦ 2 2 0 | 1  4 0 1 0  0 0 | 0 2 0  2 0 0  0 0 0  0 0 | 6 * *  *  *  *  * * * * | 1 0 1 0 0 0 0 0
... ox. xo. ...&#x ♦ 2 2 0 | 1  4 0 0 1  0 0 | 0 0 2  2 0 0  0 0 0  0 0 | * 6 *  *  *  *  * * * * | 0 1 0 0 1 0 0 0
... ... xo.3oo.&#x ♦ 3 1 0 | 3  3 0 0 0  0 0 | 1 0 0  3 0 0  0 0 0  0 0 | * * 4  *  *  *  * * * * | 1 1 0 0 0 0 0 0
oxo ... ... ...&#x ♦ 1 2 1 | 0  2 1 1 0  2 0 | 0 1 0  0 0 0  2 1 0  0 0 | * * * 12  *  *  * * * * | 0 0 1 1 0 0 0 0
... oxo ... ...&#x ♦ 1 2 1 | 0  2 1 0 1  2 0 | 0 0 1  0 0 0  2 0 1  0 0 | * * *  * 12  *  * * * * | 0 0 0 0 1 1 0 0
... ... xoo ...&#x ♦ 2 1 1 | 1  2 2 0 0  1 0 | 0 0 0  1 1 0  2 0 0  0 0 | * * *  *  * 12  * * * * | 0 0 1 0 1 0 0 0
... ... ... oox&#x ♦ 1 1 2 | 0  1 2 0 0  2 1 | 0 0 0  0 0 1  2 0 0  1 0 | * * *  *  *  * 12 * * * | 0 0 0 1 0 1 0 0
.xo ... ... .ox&#x ♦ 0 2 2 | 0  0 0 1 0  4 1 | 0 0 0  0 0 0  0 2 0  2 0 | * * *  *  *  *  * 6 * * | 0 0 0 1 0 0 1 0
... .xo ... .ox&#x ♦ 0 2 2 | 0  0 0 0 1  4 1 | 0 0 0  0 0 0  0 0 2  2 0 | * * *  *  *  *  * * 6 * | 0 0 0 0 0 1 0 1
... ... .oo3.ox&#x ♦ 0 1 3 | 0  0 0 0 0  3 3 | 0 0 0  0 0 0  0 0 0  3 1 | * * *  *  *  *  * * * 4 | 0 0 0 0 0 0 1 1
-------------------+-------+-----------------+--------------------------+-------------------------+----------------
ox. ... xo.3oo.&#x ♦ 3 2 0 | 3  6 0 1 0  0 0 | 1 3 0  6 0 0  0 0 0  0 0 | 3 0 2  0  0  0  0 0 0 0 | 2 * * * * * * *
... ox. xo.3oo.&#x ♦ 3 2 0 | 3  6 0 0 1  0 0 | 1 0 3  6 0 0  0 0 0  0 0 | 0 3 2  0  0  0  0 0 0 0 | * 2 * * * * * *
oxo ... xoo ...&#x ♦ 2 2 1 | 1  4 2 1 0  2 0 | 0 2 0  2 1 0  4 1 0  0 0 | 1 0 0  2  0  2  0 0 0 0 | * * 6 * * * * *
oxo ... ... oox&#x ♦ 1 2 2 | 0  2 2 1 0  4 1 | 0 1 0  0 0 1  4 2 0  2 0 | 0 0 0  2  0  0  2 1 0 0 | * * * 6 * * * *
... oxo xoo ...&#x ♦ 2 2 1 | 1  4 2 0 1  2 0 | 0 0 2  2 1 0  4 0 1  0 0 | 0 1 0  0  2  2  0 0 0 0 | * * * * 6 * * *
... oxo ... oox&#x ♦ 1 2 2 | 0  2 2 0 1  4 1 | 0 0 1  0 0 1  4 0 2  2 0 | 0 0 0  0  2  0  2 0 1 0 | * * * * * 6 * *
.xo ... .oo3.ox&#x ♦ 0 2 3 | 0  0 0 1 0  6 3 | 0 0 0  0 0 0  0 3 0  6 1 | 0 0 0  0  0  0  0 3 0 2 | * * * * * * 2 *
... .xo .oo3.ox&#x ♦ 0 2 3 | 0  0 0 0 1  6 3 | 0 0 0  0 0 0  0 0 3  6 1 | 0 0 0  0  0  0  0 0 3 2 | * * * * * * * 2

or
o.. o.. o..3o..    & | 6 * ♦ 2  4 2 0 0 | 1  2  2  8 3  8 |  4  4 4  4  4 12 | 2 2  6  6
.o. .o. .o.3.o.      | * 4 ♦ 0  6 0 1 1 | 0  6  6  6 0  6 |  6  6 2  6  6  6 | 2 2  6  6
---------------------+-----+------------+-----------------+------------------+----------
... ... x.. ...    & | 2 0 | 6  * * * * ♦ 1  0  0  4 1  0 |  2  2 4  0  0  4 | 2 2  2  2
oo. oo. oo.3oo.&#x & | 1 1 | * 24 * * * ♦ 0  1  1  2 0  2 |  2  2 1  2  2  3 | 1 1  3  3
o.o o.o o.o3o.o&#x   | 2 0 | *  * 6 * * ♦ 0  0  0  0 2  4 |  0  0 0  2  2  8 | 0 0  4  4
.x. ... ... ...      | 0 2 | *  * * 2 * ♦ 0  6  0  0 0  0 |  6  0 0  6  0  0 | 2 0  6  0
... .x. ... ...      | 0 2 | *  * * * 2 ♦ 0  0  6  0 0  0 |  0  6 0  0  6  0 | 0 2  0  6
---------------------+-----+------------+-----------------+------------------+----------
... ... x..3o..    & | 3 0 | 3  0 0 0 0 | 2  *  *  * *  * |  0  0 4  0  0  0 | 2 2  0  0
ox. ... ... ...&#x & | 1 2 | 0  2 0 1 0 | * 12  *  * *  * |  2  0 0  2  0  0 | 1 0  3  0
... ox. ... ...&#x & | 1 2 | 0  2 0 0 1 | *  * 12  * *  * |  0  2 0  0  2  0 | 0 1  0  3
... ... xo. ...&#x & | 2 1 | 1  2 0 0 0 | *  *  * 24 *  * |  1  1 1  0  0  1 | 1 1  1  1
... ... x.o ...&#x & | 3 0 | 1  0 2 0 0 | *  *  *  * 6  * |  0  0 0  0  0  4 | 0 0  2  2
ooo ooo ooo3ooo&#x   | 2 1 | 0  2 1 0 0 | *  *  *  * * 24 |  0  0 0  1  1  2 | 0 0  2  2
---------------------+-----+------------+-----------------+------------------+----------
ox. ... xo. ...&#x & ♦ 2 2 | 1  4 0 1 0 | 0  2  0  2 0  0 | 12  * *  *  *  * | 1 0  1  0
... ox. xo. ...&#x & ♦ 2 2 | 1  4 0 0 1 | 0  0  2  2 0  0 |  * 12 *  *  *  * | 0 1  0  1
... ... xo.3oo.&#x & ♦ 3 1 | 3  3 0 0 0 | 1  0  0  3 0  0 |  *  * 8  *  *  * | 1 1  0  0
oxo ... ... ...&#x   ♦ 2 2 | 0  4 1 1 0 | 0  2  0  0 0  2 |  *  * * 12  *  * | 0 0  2  0
... oxo ... ...&#x   ♦ 2 2 | 0  4 1 0 1 | 0  0  2  0 0  2 |  *  * *  * 12  * | 0 0  0  2
... ... xoo ...&#x & ♦ 3 1 | 1  3 2 0 0 | 0  0  0  1 1  2 |  *  * *  *  * 24 | 0 0  1  1
---------------------+-----+------------+-----------------+------------------+----------
ox. ... xo.3oo.&#x & ♦ 3 2 | 3  6 0 1 0 | 1  3  0  6 0  0 |  3  0 2  0  0  0 | 4 *  *  *
... ox. xo.3oo.&#x & ♦ 3 2 | 3  6 0 0 1 | 1  0  3  6 0  0 |  0  3 2  0  0  0 | * 4  *  *
oxo ... xoo ...&#x & ♦ 3 2 | 1  6 2 1 0 | 0  3  0  2 1  4 |  1  0 0  2  0  2 | * * 12  *
... oxo xoo ...&#x & ♦ 3 2 | 1  6 2 0 1 | 0  0  3  2 1  4 |  0  1 0  0  2  2 | * *  * 12

ox4oo xo3oo4oo&#zx   → height = 0
(tegum sum of oct and perp {4})

o.4o. o.3o.4o.     | 6 * ♦  4  4 0 | 4  4 16 | 16 16 | 16
.o4.o .o3.o4.o     | * 4 ♦  0  6 2 | 0 12 12 | 24  8 | 16
-------------------+-----+---------+---------+-------+---
.. .. x. .. ..     | 2 0 | 12  * * ♦ 2  0  4 |  4  8 |  8
oo4oo oo3oo4oo&#x  | 1 1 |  * 24 * ♦ 0  2  4 |  8  4 |  8
.x .. .. .. ..     | 0 2 |  *  * 4 ♦ 0  6  0 | 12  0 |  8
-------------------+-----+---------+---------+-------+---
.. .. x.3o. ..     | 3 0 |  3  0 0 | 8  *  * |  0  4 |  4
ox .. .. .. ..&#x  | 1 2 |  0  2 1 | * 24  * |  4  0 |  4
.. .. xo .. ..&#x  | 2 1 |  1  2 0 | *  * 48 |  2  2 |  4
-------------------+-----+---------+---------+-------+---
ox .. xo .. ..&#x  ♦ 2 2 |  1  4 1 | 0  2  2 | 48  * |  2
.. .. xo3oo ..&#x  ♦ 3 1 |  3  3 0 | 1  0  3 |  * 32 |  2
-------------------+-----+---------+---------+-------+---
ox .. xo3oo ..&#x  ♦ 3 2 |  3  6 1 | 1  3  6 |  3  2 | 32

xoo3ooo3oox oqo&#xt   → both heights = 1/sqrt(8) = 0.353553
(tet || pseudo perp q-line || dual tet)

o..3o..3o.. o..     | 4 * * ♦ 3 2  3 0 0 | 3  6  6  3  6  0 0 | 1 6 3 3 1 12  6 0 0 | 2 6  6 2 0
.o.3.o.3.o. .o.     | * 2 * ♦ 0 4  0 4 0 | 0  6  0  0 12  6 0 | 0 4 0 0 0 12 12 4 0 | 1 4  6 4 1
..o3..o3..o ..o     | * * 4 ♦ 0 0  3 2 3 | 0  0  3  6  6  6 3 | 0 0 1 3 3  6 12 6 1 | 0 2  6 6 2
--------------------+-------+------------+--------------------+---------------------+-----------
x.. ... ... ...     | 2 0 0 | 6 *  * * * ♦ 2  2  2  0  0  0 0 | 1 4 2 1 0  4  0 0 0 | 2 4  2 0 0
oo.3oo.3oo. oo.&#x  | 1 1 0 | * 8  * * * ♦ 0  3  0  0  3  0 0 | 0 3 0 0 0  6  3 0 0 | 1 3  3 1 0
o.o3o.o3o.o o.o&#x  | 1 0 1 | * * 12 * * ♦ 0  0  2  2  2  0 0 | 0 0 1 2 1  4  4 0 0 | 0 2  4 2 0
.oo3.oo3.oo .oo&#x  | 0 1 1 | * *  * 8 * ♦ 0  0  0  0  3  3 0 | 0 0 0 0 0  3  6 3 0 | 0 1  3 3 1
... ... ..x ...     | 0 0 2 | * *  * * 6 ♦ 0  0  0  2  0  2 2 | 0 0 0 1 2  0  4 4 1 | 0 0  2 4 2
--------------------+-------+------------+--------------------+---------------------+-----------
x..3o.. ... ...     | 3 0 0 | 3 0  0 0 0 | 4  *  *  *  *  * * | 1 2 1 0 0  0  0 0 0 | 2 2  0 0 0
xo. ... ... ...&#x  | 2 1 0 | 1 2  0 0 0 | * 12  *  *  *  * * | 0 2 0 0 0  2  0 0 0 | 1 2  1 0 0
x.o ... ... ...&#x  | 2 0 1 | 1 0  2 0 0 | *  * 12  *  *  * * | 0 0 1 1 0  2  0 0 0 | 0 2  2 0 0
... ... o.x ...&#x  | 1 0 2 | 0 0  2 0 1 | *  *  * 12  *  * * | 0 0 0 1 1  0  2 0 0 | 0 0  2 2 0
ooo3ooo3ooo ooo&#x  | 1 1 1 | 0 1  1 1 0 | *  *  *  * 24  * * | 0 0 0 0 0  2  2 0 0 | 0 1  2 1 0
... ... .ox ...&#x  | 0 1 2 | 0 0  0 2 1 | *  *  *  *  * 12 * | 0 0 0 0 0  0  2 2 0 | 0 0  1 2 1
... ..o3..x ...     | 0 0 3 | 0 0  0 0 3 | *  *  *  *  *  * 4 | 0 0 0 0 1  0  0 2 1 | 0 0  0 2 2
--------------------+-------+------------+--------------------+---------------------+-----------
x..3o..3o.. ...     ♦ 4 0 0 | 6 0  0 0 0 | 4  0  0  0  0  0 0 | 1 * * * *  *  * * * | 2 0  0 0 0
xo.3oo. ... ...&#x  ♦ 3 1 0 | 3 3  0 0 0 | 1  3  0  0  0  0 0 | * 8 * * *  *  * * * | 1 1  0 0 0
x.o3o.o ... ...&#x  ♦ 3 0 1 | 3 0  3 0 0 | 1  0  3  0  0  0 0 | * * 4 * *  *  * * * | 0 2  0 0 0
x.o ... o.x ...&#x  ♦ 2 0 2 | 1 0  4 0 1 | 0  0  2  2  0  0 0 | * * * 6 *  *  * * * | 0 0  2 0 0
... o.o3o.x ...&#x  ♦ 1 0 3 | 0 0  3 0 3 | 0  0  0  3  0  0 1 | * * * * 4  *  * * * | 0 0  0 2 0
xoo ... ... ...&#x  ♦ 2 1 1 | 1 2  2 1 0 | 0  1  1  0  2  0 0 | * * * * * 24  * * * | 0 1  1 0 0
... ... oox ...&#x  ♦ 1 1 2 | 0 1  2 2 1 | 0  0  0  1  2  1 0 | * * * * *  * 24 * * | 0 0  1 1 0
... .oo3.ox ...&#x  ♦ 0 1 3 | 0 0  0 3 3 | 0  0  0  0  0  3 1 | * * * * *  *  * 8 * | 0 0  0 1 1
..o3..o3..x ...     ♦ 0 0 4 | 0 0  0 0 6 | 0  0  0  0  0  0 4 | * * * * *  *  * * 1 | 0 0  0 0 2
--------------------+-------+------------+--------------------+---------------------+-----------
xo.3oo.3oo. ...&#x  ♦ 4 1 0 | 6 4  0 0 0 | 4  6  0  0  0  0 0 | 1 4 0 0 0  0  0 0 0 | 2 *  * * *
xoo3ooo ... ...&#x  ♦ 3 1 1 | 3 3  3 1 0 | 1  3  3  0  3  0 0 | 0 1 1 0 0  3  0 0 0 | * 8  * * *
xoo ... oox ...&#x  ♦ 2 1 2 | 1 2  4 2 1 | 0  1  2  2  4  1 0 | 0 0 0 1 0  2  2 0 0 | * * 12 * *
... ooo3oox ...&#x  ♦ 1 1 3 | 0 1  3 3 3 | 0  0  0  3  3  3 1 | 0 0 0 0 1  0  3 1 0 | * *  * 8 *
.oo3.oo3.ox ...&#x  ♦ 0 1 4 | 0 0  0 4 6 | 0  0  0  0  0  6 4 | 0 0 0 0 0  0  0 4 1 | * *  * * 2

oxoo3oooo3ooox&#xr   → all cyclical heights = sqrt(5/8) = 0.790569
                       in fact a rhomb with diagonals 
                       height(1,3) = sqrt(2) = 1.414214
                       height(2,4) = 1/sqrt(2) = 0.707107

o...3o...3o...    | 1 * * * ♦ 4 4 0 0  0 0 0 | 6 6 12 0 0  0  0  0 0 0 | 4 4 12 12 0 0 0 0 0  0  0 0 0 | 1 1 4 6 4 0 0 0 0 0
.o..3.o..3.o..    | * 4 * * ♦ 1 0 3 1  3 0 0 | 3 0  3 3 3  6  3  3 0 0 | 3 0  6  3 1 3 3 3 1  6  3 0 0 | 1 0 3 3 1 1 3 3 1 0
..o.3..o.3..o.    | * * 1 * ♦ 0 0 0 4  0 4 0 | 0 0  0 0 6  0  0 12 6 0 | 0 0  0  0 0 4 0 0 0 12 12 4 0 | 0 0 0 0 0 1 4 6 4 1
...o3...o3...o    | * * * 4 ♦ 0 1 0 0  3 1 3 | 0 3  3 0 0  3  6  3 3 3 | 0 3  3  6 0 0 1 3 3  3  6 3 1 | 0 1 1 3 3 0 1 3 3 1
------------------+---------+----------------+-------------------------+-------------------------------+--------------------
oo..3oo..3oo..&#x | 1 1 0 0 | 4 * * *  * * * ♦ 3 0  3 0 0  0  0  0 0 0 | 3 0  6  3 0 0 0 0 0  0  0 0 0 | 1 0 3 3 1 0 0 0 0 0
o..o3o..o3o..o&#x | 1 0 0 1 | * 4 * *  * * * ♦ 0 3  3 0 0  0  0  0 0 0 | 0 3  3  6 0 0 0 0 0  0  0 0 0 | 0 1 1 3 3 0 0 0 0 0
.x.. .... ....    | 0 2 0 0 | * * 6 *  * * * ♦ 1 0  0 2 1  2  0  0 0 0 | 2 0  2  0 1 2 2 1 0  2  0 0 0 | 1 0 2 1 0 1 2 1 0 0
.oo.3.oo.3.oo.&#x | 0 1 1 0 | * * * 4  * * * ♦ 0 0  0 0 3  0  0  3 0 0 | 0 0  0  0 0 3 0 0 0  6  3 0 0 | 0 0 0 0 0 1 3 3 1 0
.o.o3.o.o3.o.o&#x | 0 1 0 1 | * * * * 12 * * ♦ 0 0  1 0 0  2  2  1 0 0 | 0 0  2  2 0 0 1 2 1  2  2 0 0 | 0 0 1 2 1 0 1 2 1 0
..oo3..oo3..oo&#x | 0 0 1 1 | * * * *  * 4 * ♦ 0 0  0 0 0  0  0  3 3 0 | 0 0  0  0 0 0 0 0 0  3  6 3 0 | 0 0 0 0 0 0 1 3 3 1
.... .... ...x    | 0 0 0 2 | * * * *  * * 6 ♦ 0 1  0 0 0  0  2  0 1 2 | 0 2  0  2 0 0 0 1 2  0  2 2 1 | 0 1 0 1 2 0 0 1 2 1
------------------+---------+----------------+-------------------------+-------------------------------+--------------------
ox.. .... ....&#x | 1 2 0 0 | 2 0 1 0  0 0 0 | 6 *  * * *  *  *  * * * | 2 0  2  0 0 0 0 0 0  0  0 0 0 | 1 0 2 1 0 0 0 0 0 0
.... .... o..x&#x | 1 0 0 2 | 0 2 0 0  0 0 1 | * 6  * * *  *  *  * * * | 0 2  0  2 0 0 0 0 0  0  0 0 0 | 0 1 0 1 2 0 0 0 0 0
oo.o3oo.o3oo.o&#x | 1 1 0 1 | 1 1 0 0  1 0 0 | * * 12 * *  *  *  * * * | 0 0  2  2 0 0 0 0 0  0  0 0 0 | 0 0 1 2 1 0 0 0 0 0
.x..3.o.. ....    | 0 3 0 0 | 0 0 3 0  0 0 0 | * *  * 4 *  *  *  * * * | 1 0  0  0 1 1 1 0 0  0  0 0 0 | 1 0 1 0 0 1 1 0 0 0
.xo. .... ....&#x | 0 2 1 0 | 0 0 1 2  0 0 0 | * *  * * 6  *  *  * * * | 0 0  0  0 0 2 0 0 0  2  0 0 0 | 0 0 0 0 0 1 2 1 0 0
.x.o .... ....&#x | 0 2 0 1 | 0 0 1 0  2 0 0 | * *  * * * 12  *  * * * | 0 0  1  0 0 0 1 1 0  1  0 0 0 | 0 0 1 1 0 0 1 1 0 0
.... .... .o.x&#x | 0 1 0 2 | 0 0 0 0  2 0 1 | * *  * * *  * 12  * * * | 0 0  0  1 0 0 0 1 1  0  1 0 0 | 0 0 0 1 1 0 0 1 1 0
.ooo3.ooo3.ooo&#x | 0 1 1 1 | 0 0 0 1  1 1 0 | * *  * * *  *  * 12 * * | 0 0  0  0 0 0 0 0 0  2  2 0 0 | 0 0 0 0 0 0 1 2 1 0
.... .... ..ox&#x | 0 0 1 2 | 0 0 0 0  0 2 1 | * *  * * *  *  *  * 6 * | 0 0  0  0 0 0 0 0 0  0  2 2 0 | 0 0 0 0 0 0 0 1 2 1
.... ...o3...x    | 0 0 0 3 | 0 0 0 0  0 0 3 | * *  * * *  *  *  * * 4 | 0 1  0  0 0 0 0 0 1  0  0 1 1 | 0 1 0 0 1 0 0 0 1 1
------------------+---------+----------------+-------------------------+-------------------------------+--------------------
ox..3oo.. ....&#x ♦ 1 3 0 0 | 3 0 3 0  0 0 0 | 3 0  0 1 0  0  0  0 0 0 | 4 *  *  * * * * * *  *  * * * | 1 0 1 0 0 0 0 0 0 0
.... o..o3o..x&#x ♦ 1 0 0 3 | 0 3 0 0  0 0 3 | 0 3  0 0 0  0  0  0 0 1 | * 4  *  * * * * * *  *  * * * | 0 1 0 0 1 0 0 0 0 0
ox.o .... ....&#x ♦ 1 2 0 1 | 2 1 1 0  2 0 0 | 1 0  2 0 0  1  0  0 0 0 | * * 12  * * * * * *  *  * * * | 0 0 1 1 0 0 0 0 0 0
.... .... oo.x&#x ♦ 1 1 0 2 | 1 2 0 0  2 0 1 | 0 1  2 0 0  0  1  0 0 0 | * *  * 12 * * * * *  *  * * * | 0 0 0 1 1 0 0 0 0 0
.x..3.o..3.o..    ♦ 0 4 0 0 | 0 0 6 0  0 0 0 | 0 0  0 4 0  0  0  0 0 0 | * *  *  * 1 * * * *  *  * * * | 1 0 0 0 0 1 0 0 0 0
.xo.3.oo. ....&#x ♦ 0 3 1 0 | 0 0 3 3  0 0 0 | 0 0  0 1 3  0  0  0 0 0 | * *  *  * * 4 * * *  *  * * * | 0 0 0 0 0 1 1 0 0 0
.x.o3.o.o ....&#x ♦ 0 3 0 1 | 0 0 3 0  3 0 0 | 0 0  0 1 0  3  0  0 0 0 | * *  *  * * * 4 * *  *  * * * | 0 0 1 0 0 0 1 0 0 0
.x.o .... .o.x&#x ♦ 0 2 0 2 | 0 0 1 0  4 0 1 | 0 0  0 0 0  2  2  0 0 0 | * *  *  * * * * 6 *  *  * * * | 0 0 0 1 0 0 0 1 0 0
.... .o.o3.o.x&#x ♦ 0 1 0 3 | 0 0 0 0  3 0 3 | 0 0  0 0 0  0  3  0 0 1 | * *  *  * * * * * 4  *  * * * | 0 0 0 0 1 0 0 0 1 0
.xoo .... ....&#x ♦ 0 2 1 1 | 0 0 1 2  2 1 0 | 0 0  0 0 1  1  0  2 0 0 | * *  *  * * * * * * 12  * * * | 0 0 0 0 0 0 1 1 0 0
.... .... .oox&#x ♦ 0 1 1 2 | 0 0 0 1  2 2 1 | 0 0  0 0 0  0  1  2 1 0 | * *  *  * * * * * *  * 12 * * | 0 0 0 0 0 0 0 1 1 0
.... ..oo3..ox&#x ♦ 0 0 1 3 | 0 0 0 0  0 3 3 | 0 0  0 0 0  0  0  0 3 1 | * *  *  * * * * * *  *  * 4 * | 0 0 0 0 0 0 0 0 1 1
...o3...o3...x    ♦ 0 0 0 4 | 0 0 0 0  0 0 6 | 0 0  0 0 0  0  0  0 0 4 | * *  *  * * * * * *  *  * * 1 | 0 1 0 0 0 0 0 0 0 1
------------------+---------+----------------+-------------------------+-------------------------------+--------------------
ox..3oo..3oo..&#x ♦ 1 4 0 0 | 4 0 6 0  0 0 0 | 6 0  0 4 0  0  0  0 0 0 | 4 0  0  0 1 0 0 0 0  0  0 0 0 | 1 * * * * * * * * *
o..o3o..o3o..x&#x ♦ 1 0 0 4 | 0 4 0 0  0 0 6 | 0 6  0 0 0  0  0  0 0 4 | 0 4  0  0 0 0 0 0 0  0  0 0 1 | * 1 * * * * * * * *
ox.o3oo.o ....&#x ♦ 1 3 0 1 | 3 1 3 0  3 0 0 | 3 0  3 1 0  3  0  0 0 0 | 1 0  3  0 0 0 1 0 0  0  0 0 0 | * * 4 * * * * * * *
ox.o .... oo.x&#x ♦ 1 2 0 2 | 2 2 1 0  4 0 1 | 1 1  4 0 0  2  2  0 0 0 | 0 0  2  2 0 0 0 1 0  0  0 0 0 | * * * 6 * * * * * *
.... oo.o3oo.x&#x ♦ 1 1 0 3 | 1 3 0 0  3 0 3 | 0 3  3 0 0  0  3  0 0 1 | 0 1  0  3 0 0 0 0 1  0  0 0 0 | * * * * 4 * * * * *
.xo.3.oo.3.oo.&#x ♦ 0 4 1 0 | 0 0 6 4  0 0 0 | 0 0  0 4 6  0  0  0 0 0 | 0 0  0  0 1 4 0 0 0  0  0 0 0 | * * * * * 1 * * * *
.xoo3.ooo ....&#x ♦ 0 3 1 1 | 0 0 3 3  3 1 0 | 0 0  0 1 3  3  0  3 0 0 | 0 0  0  0 0 1 1 0 0  3  0 0 0 | * * * * * * 4 * * *
.xoo .... .oox&#x ♦ 0 2 1 2 | 0 0 1 2  4 2 1 | 0 0  0 0 1  2  2  4 1 0 | 0 0  0  0 0 0 0 1 0  2  2 0 0 | * * * * * * * 6 * *
.... .ooo3.oox&#x ♦ 0 1 1 3 | 0 0 0 1  3 3 3 | 0 0  0 0 0  0  3  3 3 1 | 0 0  0  0 0 0 0 0 1  0  3 1 0 | * * * * * * * * 4 *
..oo3..oo3..ox&#x ♦ 0 0 1 4 | 0 0 0 0  0 4 6 | 0 0  0 0 0  0  0  0 6 4 | 0 0  0  0 0 0 0 0 0  0  0 4 1 | * * * * * * * * * 1

or
o...3o...3o...    & | 2 * ♦  8  0  0 | 12 12 0  0 |  8 0 24 0 0 | 2  8  6
.o..3.o..3.o..    & | * 8 ♦  2  3  3 |  6  6 3  9 |  6 1 18 4 3 | 2  8  6
--------------------+-----+----------+------------+-------------+--------
oo..3oo..3oo..&#x & | 1 1 | 16  *  * ♦  3  3 0  0 |  3 0  9 0 0 | 1  4  3
.x.. .... ....    & | 0 2 |  * 12  * ♦  2  0 2  2 |  4 1  4 2 1 | 2  4  2
.o.o3.o.o3.o.o&#x   | 0 2 |  *  * 12 ♦  0  2 0  4 |  0 0  8 2 2 | 0  4  4
--------------------+-----+----------+------------+-------------+--------
ox.. .... ....&#x & | 1 2 |  2  1  0 | 24  * *  * |  2 0  2 0 0 | 1  2  1
oo.o3oo.o3oo.o&#x & | 1 2 |  2  0  1 |  * 24 *  * |  0 0  4 0 0 | 0  2  2
.x..3.o.. ....    & | 0 3 |  0  3  0 |  *  * 8  * |  2 1  0 1 0 | 2  2  0
.x.o .... ....&#x & | 0 3 |  0  1  2 |  *  * * 24 |  0 0  2 1 1 | 0  2  2
--------------------+-----+----------+------------+-------------+--------
ox..3oo.. ....&#x & ♦ 1 3 |  3  3  0 |  3  0 1  0 | 16 *  * * * | 1  1  0
.x..3.o..3.o..    & ♦ 0 4 |  0  6  0 |  0  0 4  0 |  * 2  * * * | 2  0  0
ox.o .... ....&#x & ♦ 1 3 |  3  1  2 |  1  2 0  1 |  * * 48 * * | 0  1  1
.x.o3.o.o ....&#x & ♦ 0 4 |  0  3  3 |  0  0 1  3 |  * *  * 8 * | 0  2  0
.x.o .... .o.x&#x   ♦ 0 4 |  0  2  4 |  0  0 0  4 |  * *  * * 6 | 0  0  2
--------------------+-----+----------+------------+-------------+--------
ox..3oo..3oo..&#x & ♦ 1 4 |  4  6  0 |  6  0 4  0 |  4 1  0 0 0 | 4  *  *
ox.o3oo.o ....&#x & ♦ 1 4 |  4  3  3 |  3  3 1  3 |  1 0  3 1 0 | * 16  *
ox.o .... oo.x&#x & ♦ 1 4 |  4  2  4 |  2  4 0  4 |  0 0  4 0 1 | *  * 12

o(xo)o3o(oo)o3o(ox)o&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || (tet || dual tet) || pt)

o(..).3o(..).3o(..).     & | 2 * ♦ 4 4  0  0 |  6  6 12 0  0 | 4 4 12 12 0 0 0 | 1 1 4  6 4
.(o.).3.(o.).3.(o.).     & | * 8 ♦ 1 1  3  3 |  3  3  6 3  9 | 3 3  9  9 1 4 3 | 1 1 4  6 4
---------------------------+-----+-----------+---------------+-----------------+-----------
o(o.).3o(o.).3o(o.).&#x  & | 1 1 | 8 *  *  * ♦  3  0  3 0  0 | 3 0  6  3 0 0 0 | 1 0 3  3 1
o(.o).3o(.o).3o(.o).&#x  & | 1 1 | * 8  *  * ♦  0  3  3 0  0 | 0 3  3  6 0 0 0 | 0 1 1  3 3
.(x.). .(..). .(..).     & | 0 2 | * * 12  * ♦  1  1  0 2  2 | 2 2  2  2 1 2 1 | 1 1 2  2 2
.(oo).3.(oo).3.(oo).&#x    | 0 2 | * *  * 12 ♦  0  0  2 0  4 | 0 0  4  4 0 2 2 | 0 0 2  4 2
---------------------------+-----+-----------+---------------+-----------------+-----------
o(x.). .(..). .(..).&#x  & | 1 2 | 2 0  1  0 | 12  *  * *  * | 2 0  2  0 0 0 0 | 1 0 2  1 0
.(..). .(..). o(.x).&#x  & | 1 2 | 0 2  1  0 |  * 12  * *  * | 0 2  0  2 0 0 0 | 0 1 0  1 2
o(oo).3o(oo).3o(oo).&#x  & | 1 2 | 1 1  0  1 |  *  * 24 *  * | 0 0  2  2 0 0 0 | 0 0 1  2 1
.(x.).3.(o.). .(..).     & | 0 3 | 0 0  3  0 |  *  *  * 8  * | 1 1  0  0 1 1 0 | 1 1 1  0 1
.(xo). .(..). .(..).&#x  & | 0 3 | 0 0  1  2 |  *  *  * * 24 | 0 0  1  1 0 1 1 | 0 0 1  2 1
---------------------------+-----+-----------+---------------+-----------------+-----------
o(x.).3o(o.). .(..).&#x  & ♦ 1 3 | 3 0  3  0 |  3  0  0 1  0 | 8 *  *  * * * * | 1 0 1  0 0
.(..). o(.o).3o(.x).&#x  & ♦ 1 3 | 0 3  3  0 |  0  3  0 1  0 | * 8  *  * * * * | 0 1 0  0 1
o(xo). .(..). .(..).&#x  & ♦ 1 3 | 2 1  1  2 |  1  0  2 0  1 | * * 24  * * * * | 0 0 1  1 0
.(..). .(..). o(ox).&#x  & ♦ 1 3 | 1 2  1  2 |  0  1  2 0  1 | * *  * 24 * * * | 0 0 0  1 1
.(x.).3.(o.).3.(o.).     & ♦ 0 4 | 0 0  6  0 |  0  0  0 4  0 | * *  *  * 2 * * | 1 1 0  0 0
.(xo).3.(oo). .(..).&#x  & ♦ 0 4 | 0 0  3  3 |  0  0  0 1  3 | * *  *  * * 8 * | 0 0 1  0 1
.(xo). .(..). .(ox).&#x    ♦ 0 4 | 0 0  2  4 |  0  0  0 0  4 | * *  *  * * * 6 | 0 0 0  2 0
---------------------------+-----+-----------+---------------+-----------------+-----------
o(x.).3o(o.).3o(o.).&#x  & ♦ 1 4 | 4 0  6  0 |  6  0  0 4  0 | 4 0  0  0 1 0 0 | 2 * *  * *
o(.o).3o(.o).3o(.x).&#x  & ♦ 1 4 | 0 4  6  0 |  0  6  0 4  0 | 0 4  0  0 1 0 0 | * 2 *  * *
o(xo).3o(oo). .(..).&#x  & ♦ 1 4 | 3 1  3  3 |  3  0  3 1  3 | 1 0  3  0 0 1 0 | * * 8  * *
o(xo). .(..). o(ox).&#x  & ♦ 1 4 | 2 2  2  4 |  1  1  4 0  4 | 0 0  2  2 0 0 1 | * * * 12 *
.(..). o(oo).3o(ox).&#x  & ♦ 1 4 | 1 3  3  3 |  0  3  3 1  3 | 0 1  0  3 0 1 0 | * * *  * 8

xoxo oxoo3ooox&#xr   → all cyclical heights = sqrt(5/12) = 0.645497
                       in fact a rhomb with diagonals 
                       height(1,3) = 1
                       height(2,4) = sqrt(2/3) = 0.816497

o... o...3o...    | 2 * * * ♦ 1 3 1 3 0 0 0 0 0 0 | 3 3 3 3 3  6 3 0 0 0 0 0  0 0 0 0 | 3 1 3 1 3 6 3 3 3  6 0 0 0 0 0 0 0 | 1 1 1 3 3 1 3 3 0 0 0 0
.o.. .o..3.o..    | * 3 * * ♦ 0 2 0 0 2 2 2 0 0 0 | 1 4 0 0 2  4 0 1 1 4 2 1  4 0 0 0 | 2 2 0 0 4 2 4 2 0  4 2 2 2 4 2 0 0 | 1 0 2 2 1 0 4 2 1 2 1 0
..o. ..o.3..o.    | * * 2 * ♦ 0 0 1 0 0 3 0 1 3 0 | 0 0 0 0 3  0 3 0 3 3 0 0  6 3 3 0 | 0 0 0 0 3 0 0 0 3  6 3 1 6 3 3 3 1 | 0 0 1 0 0 1 3 3 1 3 3 1
...o ...o3...o    | * * * 3 ♦ 0 0 0 2 0 0 2 0 2 2 | 0 0 1 4 0  4 2 0 0 0 1 2  4 1 4 1 | 0 0 2 2 0 2 2 4 4  4 0 0 2 2 4 2 2 | 0 1 0 1 2 2 2 4 0 1 2 1
------------------+---------+---------------------+-----------------------------------+------------------------------------+------------------------
x... .... ....    | 2 0 0 0 | 1 * * * * * * * * * ♦ 3 0 3 0 0  0 0 0 0 0 0 0  0 0 0 0 | 3 0 3 0 0 6 0 0 0  0 0 0 0 0 0 0 0 | 1 1 0 3 3 0 0 0 0 0 0 0
oo.. oo..3oo..&#x | 1 1 0 0 | * 6 * * * * * * * * ♦ 1 2 0 0 1  2 0 0 0 0 0 0  0 0 0 0 | 2 1 0 0 2 2 2 1 0  2 0 0 0 0 0 0 0 | 1 0 1 2 1 0 2 1 0 0 0 0
o.o. o.o.3o.o.&#x | 1 0 1 0 | * * 2 * * * * * * * ♦ 0 0 0 0 3  0 3 0 0 0 0 0  0 0 0 0 | 0 0 0 0 3 0 0 0 3  6 0 0 0 0 0 0 0 | 0 0 1 0 0 1 3 3 0 0 0 0
o..o o..o3o..o&#x | 1 0 0 1 | * * * 6 * * * * * * ♦ 0 0 1 2 0  2 1 0 0 0 0 0  0 0 0 0 | 0 0 2 1 0 2 1 2 2  2 0 0 0 0 0 0 0 | 0 1 0 1 2 1 1 2 0 0 0 0
.... .x.. ....    | 0 2 0 0 | * * * * 3 * * * * * ♦ 0 2 0 0 0  0 0 1 0 2 1 0  0 0 0 0 | 1 2 0 0 2 0 2 0 0  0 1 2 0 2 0 0 0 | 1 0 2 1 0 0 2 0 1 1 0 0
.oo. .oo.3.oo.&#x | 0 1 1 0 | * * * * * 6 * * * * ♦ 0 0 0 0 1  0 0 0 1 2 0 0  2 0 0 0 | 0 0 0 0 2 0 0 0 0  2 2 1 2 2 1 0 0 | 0 0 1 0 0 0 2 1 1 2 1 0
.o.o .o.o3.o.o&#x | 0 1 0 1 | * * * * * * 6 * * * ♦ 0 0 0 0 0  2 0 0 0 0 1 1  2 0 0 0 | 0 0 0 0 0 1 2 2 0  2 0 0 1 2 2 0 0 | 0 0 0 1 1 0 2 2 0 1 1 0
..x. .... ....    | 0 0 2 0 | * * * * * * * 1 * * ♦ 0 0 0 0 0  0 0 0 3 0 0 0  0 3 0 0 | 0 0 0 0 0 0 0 0 0  0 3 0 6 0 0 3 0 | 0 0 0 0 0 0 0 0 1 3 3 1
..oo ..oo3..oo&#x | 0 0 1 1 | * * * * * * * * 6 * ♦ 0 0 0 0 0  0 1 0 0 0 0 0  2 1 2 0 | 0 0 0 0 0 0 0 0 2  2 0 0 2 1 2 2 1 | 0 0 0 0 0 1 1 2 0 1 2 1
.... .... ...x    | 0 0 0 2 | * * * * * * * * * 3 ♦ 0 0 0 2 0  0 0 0 0 0 0 1  0 0 2 1 | 0 0 1 2 0 0 0 2 2  0 0 0 0 0 2 1 2 | 0 1 0 0 1 2 0 2 0 0 1 1
------------------+---------+---------------------+-----------------------------------+------------------------------------+------------------------
xo.. .... ....&#x | 2 1 0 0 | 1 2 0 0 0 0 0 0 0 0 | 3 * * * *  * * * * * * *  * * * * | 2 0 0 0 0 2 0 0 0  0 0 0 0 0 0 0 0 | 1 0 0 2 1 0 0 0 0 0 0 0
.... ox.. ....&#x | 1 2 0 0 | 0 2 0 0 1 0 0 0 0 0 | * 6 * * *  * * * * * * *  * * * * | 1 1 0 0 1 0 1 0 0  0 0 0 0 0 0 0 0 | 1 0 1 1 0 0 1 0 0 0 0 0
x..o .... ....&#x | 2 0 0 1 | 1 0 0 2 0 0 0 0 0 0 | * * 3 * *  * * * * * * *  * * * * | 0 0 2 0 0 2 0 0 0  0 0 0 0 0 0 0 0 | 0 1 0 1 2 0 0 0 0 0 0 0
.... .... o..x&#x | 1 0 0 2 | 0 0 0 2 0 0 0 0 0 1 | * * * 6 *  * * * * * * *  * * * * | 0 0 1 1 0 0 0 1 1  0 0 0 0 0 0 0 0 | 0 1 0 0 1 1 0 1 0 0 0 0
ooo. ooo.3ooo.&#x | 1 1 1 0 | 0 1 1 0 0 1 0 0 0 0 | * * * * 6  * * * * * * *  * * * * | 0 0 0 0 2 0 0 0 0  2 0 0 0 0 0 0 0 | 0 0 1 0 0 0 2 1 0 0 0 0
oo.o oo.o3oo.o&#x | 1 1 0 1 | 0 1 0 1 0 0 1 0 0 0 | * * * * * 12 * * * * * *  * * * * | 0 0 0 0 0 1 1 1 0  1 0 0 0 0 0 0 0 | 0 0 0 1 1 0 1 1 0 0 0 0
o.oo o.oo3o.oo&#x | 1 0 1 1 | 0 0 1 1 0 0 0 0 1 0 | * * * * *  * 6 * * * * *  * * * * | 0 0 0 0 0 0 0 0 2  2 0 0 0 0 0 0 0 | 0 0 0 0 0 1 1 2 0 0 0 0
.... .x..3.o..    | 0 3 0 0 | 0 0 0 0 3 0 0 0 0 0 | * * * * *  * * 1 * * * *  * * * * | 0 2 0 0 0 0 0 0 0  0 0 2 0 0 0 0 0 | 1 0 2 0 0 0 0 0 1 0 0 0
.ox. .... ....&#x | 0 1 2 0 | 0 0 0 0 0 2 0 1 0 0 | * * * * *  * * * 3 * * *  * * * * | 0 0 0 0 0 0 0 0 0  0 2 0 2 0 0 0 0 | 0 0 0 0 0 0 0 0 1 2 1 0
.... .xo. ....&#x | 0 2 1 0 | 0 0 0 0 1 2 0 0 0 0 | * * * * *  * * * * 6 * *  * * * * | 0 0 0 0 1 0 0 0 0  0 1 1 0 1 0 0 0 | 0 0 1 0 0 0 1 0 1 1 0 0
.... .x.o ....&#x | 0 2 0 1 | 0 0 0 0 1 0 2 0 0 0 | * * * * *  * * * * * 3 *  * * * * | 0 0 0 0 0 0 2 0 0  0 0 0 0 2 0 0 0 | 0 0 0 1 0 0 2 0 0 1 0 0
.... .... .o.x&#x | 0 1 0 2 | 0 0 0 0 0 0 2 0 0 1 | * * * * *  * * * * * * 3  * * * * | 0 0 0 0 0 0 0 2 0  0 0 0 0 0 2 0 0 | 0 0 0 0 1 0 0 2 0 0 1 0
.ooo .ooo3.ooo&#x | 0 1 1 1 | 0 0 0 0 0 1 1 0 1 0 | * * * * *  * * * * * * * 12 * * * | 0 0 0 0 0 0 0 0 0  1 0 0 1 1 1 0 0 | 0 0 0 0 0 0 1 1 0 1 1 0
..xo .... ....&#x | 0 0 2 1 | 0 0 0 0 0 0 0 1 2 0 | * * * * *  * * * * * * *  * 3 * * | 0 0 0 0 0 0 0 0 0  0 0 0 2 0 0 2 0 | 0 0 0 0 0 0 0 0 0 1 2 1
.... .... ..ox&#x | 0 0 1 2 | 0 0 0 0 0 0 0 0 2 1 | * * * * *  * * * * * * *  * * 6 * | 0 0 0 0 0 0 0 0 1  0 0 0 0 0 1 1 1 | 0 0 0 0 0 1 0 1 0 0 1 1
.... ...o3...x    | 0 0 0 3 | 0 0 0 0 0 0 0 0 0 3 | * * * * *  * * * * * * *  * * * 1 | 0 0 0 2 0 0 0 0 0  0 0 0 0 0 0 0 2 | 0 1 0 0 0 2 0 0 0 0 0 1
------------------+---------+---------------------+-----------------------------------+------------------------------------+------------------------
xo.. ox.. ....&#x ♦ 2 2 0 0 | 1 4 0 0 1 0 0 0 0 0 | 2 2 0 0 0  0 0 0 0 0 0 0  0 0 0 0 | 3 * * * * * * * *  * * * * * * * * | 1 0 0 1 0 0 0 0 0 0 0 0
.... ox..3oo..&#x ♦ 1 3 0 0 | 0 3 0 0 3 0 0 0 0 0 | 0 3 0 0 0  0 0 1 0 0 0 0  0 0 0 0 | * 2 * * * * * * *  * * * * * * * * | 1 0 1 0 0 0 0 0 0 0 0 0
x..o .... o..x&#x ♦ 2 0 0 2 | 1 0 0 4 0 0 0 0 0 1 | 0 0 2 2 0  0 0 0 0 0 0 0  0 0 0 0 | * * 3 * * * * * *  * * * * * * * * | 0 1 0 0 1 0 0 0 0 0 0 0
.... o..o3o..x&#x ♦ 1 0 0 3 | 0 0 0 3 0 0 0 0 0 3 | 0 0 0 3 0  0 0 0 0 0 0 0  0 0 0 1 | * * * 2 * * * * *  * * * * * * * * | 0 1 0 0 0 1 0 0 0 0 0 0
.... oxo. ....&#x ♦ 1 2 1 0 | 0 2 1 0 1 2 0 0 0 0 | 0 1 0 0 2  0 0 0 0 1 0 0  0 0 0 0 | * * * * 6 * * * *  * * * * * * * * | 0 0 1 0 0 0 1 0 0 0 0 0
xo.o .... ....&#x ♦ 2 1 0 1 | 1 2 0 2 0 0 1 0 0 0 | 1 0 1 0 0  2 0 0 0 0 0 0  0 0 0 0 | * * * * * 6 * * *  * * * * * * * * | 0 0 0 1 1 0 0 0 0 0 0 0
.... ox.o ....&#x ♦ 1 2 0 1 | 0 2 0 1 1 0 2 0 0 0 | 0 1 0 0 0  2 0 0 0 0 1 0  0 0 0 0 | * * * * * * 6 * *  * * * * * * * * | 0 0 0 1 0 0 1 0 0 0 0 0
.... .... oo.x&#x ♦ 1 1 0 2 | 0 1 0 2 0 0 2 0 0 1 | 0 0 0 1 0  2 0 0 0 0 0 1  0 0 0 0 | * * * * * * * 6 *  * * * * * * * * | 0 0 0 0 1 0 0 1 0 0 0 0
.... .... o.ox&#x ♦ 1 0 1 2 | 0 0 1 2 0 0 0 0 2 1 | 0 0 0 1 0  0 2 0 0 0 0 0  0 0 1 0 | * * * * * * * * 6  * * * * * * * * | 0 0 0 0 0 1 0 1 0 0 0 0
oooo oooo3oooo&#x ♦ 1 1 1 1 | 0 1 1 1 0 1 1 0 1 0 | 0 0 0 0 1  1 1 0 0 0 0 0  1 0 0 0 | * * * * * * * * * 12 * * * * * * * | 0 0 0 0 0 0 1 1 0 0 0 0
.ox. .xo. ....&#x ♦ 0 2 2 0 | 0 0 0 0 1 4 0 1 0 0 | 0 0 0 0 0  0 0 0 2 2 0 0  0 0 0 0 | * * * * * * * * *  * 3 * * * * * * | 0 0 0 0 0 0 0 0 1 1 0 0
.... .xo.3.oo.&#x ♦ 0 3 1 0 | 0 0 0 0 3 3 0 0 0 0 | 0 0 0 0 0  0 0 1 0 3 0 0  0 0 0 0 | * * * * * * * * *  * * 2 * * * * * | 0 0 1 0 0 0 0 0 1 0 0 0
.oxo .... ....&#x ♦ 0 1 2 1 | 0 0 0 0 0 2 1 1 2 0 | 0 0 0 0 0  0 0 0 1 0 0 0  2 1 0 0 | * * * * * * * * *  * * * 6 * * * * | 0 0 0 0 0 0 0 0 0 1 1 0
.... .xoo ....&#x ♦ 0 2 1 1 | 0 0 0 0 1 2 2 0 1 0 | 0 0 0 0 0  0 0 0 0 1 1 0  2 0 0 0 | * * * * * * * * *  * * * * 6 * * * | 0 0 0 0 0 0 1 0 0 1 0 0
.... .... .oox&#x ♦ 0 1 1 2 | 0 0 0 0 0 1 2 0 2 1 | 0 0 0 0 0  0 0 0 0 0 0 1  2 0 1 0 | * * * * * * * * *  * * * * * 6 * * | 0 0 0 0 0 0 0 1 0 0 1 0
..xo .... ..ox&#x ♦ 0 0 2 2 | 0 0 0 0 0 0 0 1 4 1 | 0 0 0 0 0  0 0 0 0 0 0 0  0 2 2 0 | * * * * * * * * *  * * * * * * 3 * | 0 0 0 0 0 0 0 0 0 0 1 1
.... ..oo3..ox&#x ♦ 0 0 1 3 | 0 0 0 0 0 0 0 0 3 3 | 0 0 0 0 0  0 0 0 0 0 0 0  0 0 3 1 | * * * * * * * * *  * * * * * * * 2 | 0 0 0 0 0 1 0 0 0 0 0 1
------------------+---------+---------------------+-----------------------------------+------------------------------------+------------------------
xo.. ox..3oo..&#x ♦ 2 3 0 0 | 1 6 0 0 3 0 0 0 0 0 | 3 6 0 0 0  0 0 1 0 0 0 0  0 0 0 0 | 3 2 0 0 0 0 0 0 0  0 0 0 0 0 0 0 0 | 1 * * * * * * * * * * *
x..o o..o3o..x&#x ♦ 2 0 0 3 | 1 0 0 6 0 0 0 0 0 3 | 0 0 3 6 0  0 0 0 0 0 0 0  0 0 0 1 | 0 0 3 2 0 0 0 0 0  0 0 0 0 0 0 0 0 | * 1 * * * * * * * * * *
.... oxo.3ooo.&#x ♦ 1 3 1 0 | 0 3 1 0 3 3 0 0 0 0 | 0 3 0 0 3  0 0 1 0 3 0 0  0 0 0 0 | 0 1 0 0 3 0 0 0 0  0 0 1 0 0 0 0 0 | * * 2 * * * * * * * * *
xo.o ox.o ....&#x ♦ 2 2 0 1 | 1 4 0 2 1 0 2 0 0 0 | 2 2 1 0 0  4 0 0 0 0 1 0  0 0 0 0 | 1 0 0 0 0 2 2 0 0  0 0 0 0 0 0 0 0 | * * * 3 * * * * * * * *
xo.o .... oo.x&#x ♦ 2 1 0 2 | 1 2 0 4 0 0 2 0 0 1 | 1 0 2 2 0  4 0 0 0 0 0 1  0 0 0 0 | 0 0 1 0 0 2 0 2 0  0 0 0 0 0 0 0 0 | * * * * 3 * * * * * * *
.... o.oo3o.ox&#x ♦ 1 0 1 3 | 0 0 1 3 0 0 0 0 3 3 | 0 0 0 3 0  0 3 0 0 0 0 0  0 0 3 1 | 0 0 0 1 0 0 0 0 3  0 0 0 0 0 0 0 1 | * * * * * 2 * * * * * *
.... oxoo ....&#x ♦ 1 2 1 1 | 0 2 1 1 1 2 2 0 1 0 | 0 1 0 0 2  2 1 0 0 1 1 0  2 0 0 0 | 0 0 0 0 1 0 1 0 0  2 0 0 0 1 0 0 0 | * * * * * * 6 * * * * *
.... .... ooox&#x ♦ 1 1 1 2 | 0 1 1 2 0 1 2 0 2 1 | 0 0 0 1 1  2 2 0 0 0 0 1  2 0 1 0 | 0 0 0 0 0 0 0 1 1  2 0 0 0 0 1 0 0 | * * * * * * * 6 * * * *
.ox. .xo.3.oo.&#x ♦ 0 3 2 0 | 0 0 0 0 3 6 0 1 0 0 | 0 0 0 0 0  0 0 1 3 6 0 0  0 0 0 0 | 0 0 0 0 0 0 0 0 0  0 3 2 0 0 0 0 0 | * * * * * * * * 1 * * *
.oxo .xoo ....&#x ♦ 0 2 2 1 | 0 0 0 0 1 4 2 1 2 0 | 0 0 0 0 0  0 0 0 2 2 1 0  4 1 0 0 | 0 0 0 0 0 0 0 0 0  0 1 0 2 2 0 0 0 | * * * * * * * * * 3 * *
.oxo .... .oox&#x ♦ 0 1 2 2 | 0 0 0 0 0 2 2 1 4 1 | 0 0 0 0 0  0 0 0 1 0 0 1  4 2 2 0 | 0 0 0 0 0 0 0 0 0  0 0 0 2 0 2 1 0 | * * * * * * * * * * 3 *
..xo ..oo3..ox&#x ♦ 0 0 2 3 | 0 0 0 0 0 0 0 1 6 3 | 0 0 0 0 0  0 0 0 0 0 0 0  0 3 6 1 | 0 0 0 0 0 0 0 0 0  0 0 0 0 0 0 3 2 | * * * * * * * * * * * 1

xo4oo ox3oo4oo&#zx   → height = 0
(tegum sum of {4} and perp oct)
(tegum product of {4} with oct)

o.4o. o.3o.4o.    | 4 * ♦ 2  6  0 | 12 12 0 | 24  8 | 16
.o4.o .o3.o4.o    | * 6 ♦ 0  4  4 |  4 16 4 | 16 16 | 16
------------------+-----+---------+---------+-------+---
x. .. .. .. ..    | 2 0 | 4  *  * ♦  6  0 0 | 12  0 |  8
oo4oo oo3oo4oo&#x | 1 1 | * 24  * ♦  2  4 0 |  8  4 |  8
.. .. .x .. ..    | 0 2 | *  * 12 ♦  0  4 2 |  4  8 |  8
------------------+-----+---------+---------+-------+---
xo .. .. .. ..&#x | 2 1 | 1  2  0 | 24  * * |  4  0 |  4
.. .. ox .. ..&#x | 1 2 | 0  2  1 |  * 48 * |  2  2 |  4
.. .. .x3.o ..    | 0 3 | 0  0  3 |  *  * 8 |  0  4 |  4
------------------+-----+---------+---------+-------+---
xo .. ox .. ..&#x ♦ 2 2 | 1  4  1 |  2  2 0 | 48  * |  2
.. .. ox3oo ..&#x ♦ 1 3 | 0  3  3 |  0  3 1 |  * 32 |  2
------------------+-----+---------+---------+-------+---
xo .. ox3oo ..&#x ♦ 2 3 | 1  6  3 |  3  6 1 |  3  2 | 32

xo xo oo3ox3oo&#zx   → height = 0
(tegum sum of {4} and perp oct)
(tegum product of {4} with oct)

o. o. o.3o.3o.    | 4 * ♦ 1 1  6  0 |  6  6 12 0 0 | 12 12  4  4 | 4 4 4 4
.o .o .o3.o3.o    | * 6 ♦ 0 0  4  4 |  2  2 16 2 2 |  8  8  8  8 | 4 4 4 4
------------------+-----+-----------+--------------+-------------+--------
x. .. .. .. ..    | 2 0 | 2 *  *  * ♦  6  0  0 0 0 | 12  0  0  0 | 4 4 0 0
.. x. .. .. ..    | 2 0 | * 2  *  * ♦  0  6  0 0 0 |  0 12  0  0 | 0 0 4 4
oo oo oo3oo3oo&#x | 1 1 | * * 24  * ♦  1  1  4 0 0 |  4  4  2  2 | 2 2 2 2
.. .. .. .x ..    | 0 2 | * *  * 12 ♦  0  0  4 1 1 |  2  2  4  4 | 2 2 2 2
------------------+-----+-----------+--------------+-------------+--------
xo .. .. .. ..&#x | 2 1 | 1 0  2  0 | 12  *  * * * |  4  0  0  0 | 2 2 0 0
.. xo .. .. ..&#x | 2 1 | 0 1  2  0 |  * 12  * * * |  0  4  0  0 | 0 0 2 2
.. .. .. ox ..&#x | 1 2 | 0 0  2  1 |  *  * 48 * * |  1  1  1  1 | 1 1 1 1
.. .. .o3.x ..    | 0 3 | 0 0  0  3 |  *  *  * 4 * |  0  0  4  0 | 2 0 2 0
.. .. .. .x3.o    | 0 3 | 0 0  0  3 |  *  *  * * 4 |  0  0  0  4 | 0 2 0 2
------------------+-----+-----------+--------------+-------------+--------
xo .. .. ox ..&#x ♦ 2 2 | 1 0  4  1 |  2  0  2 0 0 | 24  *  *  * | 1 1 0 0
.. xo .. ox ..&#x ♦ 2 2 | 0 1  4  1 |  0  2  2 0 0 |  * 24  *  * | 0 0 1 1
.. .. oo3ox ..&#x ♦ 1 3 | 0 0  3  3 |  0  0  3 1 0 |  *  * 16  * | 1 0 1 0
.. .. .. ox3oo&#x ♦ 1 3 | 0 0  3  3 |  0  0  3 0 1 |  *  *  * 16 | 0 1 0 1
------------------+-----+-----------+--------------+-------------+--------
xo .. oo3ox ..&#x ♦ 2 3 | 1 0  6  3 |  3  0  6 1 0 |  3  0  2  0 | 8 * * *
xo .. .. ox3oo&#x ♦ 2 3 | 1 0  6  3 |  3  0  6 0 1 |  3  0  0  2 | * 8 * *
.. xo oo3ox ..&#x ♦ 2 3 | 0 1  6  3 |  0  3  6 1 0 |  0  3  2  0 | * * 8 *
.. xo .. ox3oo&#x ♦ 2 3 | 0 1  6  3 |  0  3  6 0 1 |  0  3  0  2 | * * * 8

o(xo)o4o(oo)o o(ox)o4o(oo)o&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || (tegum product of two {4}) || pt)

o(..).4o(..). o(..).4o(..).    | 1 * * * ♦ 4 4 0  0 0 0 0 | 4 16 4  0  0 0  0 0 | 16 16  0  0  0 | 16  0
.(o.).4.(o.). .(o.).4.(o.).    | * 4 * * ♦ 1 0 2  4 1 0 0 | 2  4 0  8  4 2  4 0 |  8  4  8  8  4 |  8  8
.(.o).4.(.o). .(.o).4.(.o).    | * * 4 * ♦ 0 1 0  4 0 2 1 | 0  4 2  4  8 0  4 2 |  4  8  8  4  8 |  8  8
.(..)o4.(..)o .(..)o4.(..)o    | * * * 1 ♦ 0 0 0  0 4 0 4 | 0  0 0  0  0 4 16 4 |  0  0  0 16 16 |  0 16
-------------------------------+---------+----------------+---------------------+----------------+------
o(o.).4o(o.). o(o.).4o(o.).&#x | 1 1 0 0 | 4 * *  * * * * ♦ 2  4 0  0  0 0  0 0 |  8  4  0  0  0 |  8  0
o(.o).4o(.o). o(.o).4o(.o).&#x | 1 0 1 0 | * 4 *  * * * * ♦ 0  4 2  0  0 0  0 0 |  4  8  0  0  0 |  4  0
.(x.). .(..). .(..). .(..).    | 0 2 0 0 | * * 4  * * * * ♦ 1  0 0  4  0 1  0 0 |  4  0  4  4  0 |  4  4
.(oo).4.(oo). .(oo).4.(oo).&#x | 0 1 1 0 | * * * 16 * * * ♦ 0  1 0  2  2 0  1 0 |  2  2  4  2  2 |  4  4
.(o.)o4.(o.)o .(o.)o4.(o.)o&#x | 0 1 0 1 | * * *  * 4 * * ♦ 0  0 0  0  0 2  4 0 |  0  0  0  8  4 |  0  8
.(..). .(..). .(.x). .(..).    | 0 0 2 0 | * * *  * * 4 * ♦ 0  0 1  0  4 0  0 1 |  0  4  4  0  4 |  4  4
.(.o)o4.(.o)o .(.o)o4.(.o)o&#x | 0 0 1 1 | * * *  * * * 4 ♦ 0  0 0  0  0 0  4 2 |  0  0  0  4  8 |  0  8
-------------------------------+---------+----------------+---------------------+----------------+------
o(x.). .(..). .(..). .(..).&#x | 1 2 0 0 | 2 0 1  0 0 0 0 | 4  * *  *  * *  * * |  4  0  0  0  0 |  4  0
o(oo).4o(oo). o(oo).4o(oo).&#x | 1 1 1 0 | 1 1 0  1 0 0 0 | * 16 *  *  * *  * * |  2  2  0  0  0 |  4  0
.(..). .(..). o(.x). .(..).&#x | 1 0 2 0 | 0 2 0  0 0 1 0 | *  * 4  *  * *  * * |  0  4  0  0  0 |  4  0
.(xo). .(..). .(..). .(..).&#x | 0 2 1 0 | 0 0 1  2 0 0 0 | *  * * 16  * *  * * |  1  0  2  1  0 |  2  2
.(..). .(..). .(ox). .(..).&#x | 0 1 2 0 | 0 0 0  2 0 1 0 | *  * *  * 16 *  * * |  0  1  2  0  1 |  2  2
.(x.)o .(..). .(..). .(..).&#x | 0 2 0 1 | 0 0 1  0 2 0 0 | *  * *  *  * 4  * * |  0  0  0  4  0 |  0  4
.(oo)o4.(oo)o .(oo)o4.(oo)o&#x | 0 1 1 1 | 0 0 0  1 1 0 1 | *  * *  *  * * 16 * |  0  0  0  2  2 |  0  4
.(..). .(..). .(.x)o .(..).&#x | 0 0 2 1 | 0 0 0  0 0 1 2 | *  * *  *  * *  * 4 |  0  0  0  0  4 |  0  4
-------------------------------+---------+----------------+---------------------+----------------+------
o(xo). .(..). .(..). .(..).&#x ♦ 1 2 1 0 | 2 1 1  2 0 0 0 | 1  2 0  1  0 0  0 0 | 16  *  *  *  * |  2  0
.(..). .(..). o(ox). .(..).&#x ♦ 1 1 2 0 | 1 2 0  2 0 1 0 | 0  2 1  0  1 0  0 0 |  * 16  *  *  * |  2  0
.(xo). .(..). .(ox). .(..).&#x ♦ 0 2 2 0 | 0 0 1  4 0 1 0 | 0  0 0  2  2 0  0 0 |  *  * 16  *  * |  1  1
.(xo)o .(..). .(..). .(..).&#x ♦ 0 2 1 1 | 0 0 1  2 2 0 1 | 0  0 0  1  0 1  2 0 |  *  *  * 16  * |  0  2
.(..). .(..). .(ox)o .(..).&#x ♦ 0 1 2 1 | 0 0 0  2 1 1 2 | 0  0 0  0  1 0  2 1 |  *  *  *  * 16 |  0  2
-------------------------------+---------+----------------+---------------------+----------------+------
o(xo). .(..). o(ox). .(..).&#x ♦ 1 2 2 0 | 2 2 1  4 0 1 0 | 1  4 1  2  2 0  0 0 |  2  2  1  0  0 | 16  *
.(xo)o .(..). .(ox)o .(..).&#x ♦ 0 2 2 1 | 0 0 1  4 2 1 2 | 0  0 0  2  2 1  4 1 |  0  0  1  2  2 |  * 16

ooq xoo4ooo oxo4ooo&#zx   → heights = 0
(tegum sum of 2 perp {4} and a perp q-line)

o.. o..4o.. o..4o..     & | 8 * ♦ 2  4  2 | 12  4  8 |  8 24 | 16
..o ..o4..o ..o4..o       | * 2 ♦ 0  0  8 |  0  8 16 |  0 32 | 16
--------------------------+-----+---------+----------+-------+---
... x.. ... ... ...     & | 2 0 | 8  *  * ♦  4  2  0 |  4  8 |  8
oo. oo.4oo. oo.4oo.&#x    | 2 0 | * 16  * ♦  4  0  2 |  4  8 |  8
o.o o.o4o.o o.o4o.o&#x  & | 1 1 | *  * 16 ♦  0  2  4 |  0 12 |  8
--------------------------+-----+---------+----------+-------+---
... xo. ... ... ...&#x  & | 3 0 | 1  2  0 | 32  *  * |  2  2 |  4
... x.o ... ... ...&#x  & | 2 1 | 1  0  2 |  * 16  * |  0  4 |  4
ooo ooo4ooo ooo4ooo&#x    | 2 1 | 0  1  2 |  *  * 32 |  0  4 |  4
--------------------------+-----+---------+----------+-------+---
... xo. ... ox. ...&#x    ♦ 4 0 | 2  4  0 |  4  0  0 | 16  * |  2
... xoo ... ... ...&#x  & ♦ 3 1 | 1  2  3 |  1  1  2 |  * 64 |  2
--------------------------+-----+---------+----------+-------+---
... xoo ... oxo ...&#x    ♦ 4 1 | 2  4  4 |  4  2  4 |  1  4 | 32

o(xo)o o(xo)o o(ox)o o(ox)o&#xt   → both heights = 1/sqrt(2) = 0.707107
(pt || (tegum product of two {4}) || pt)

o(..). o(..). o(..). o(..).    | 1 * * * ♦ 4 4 0 0  0 0 0 0 0 | 2 2 16 2 2 0 0 0 0 0 0  0 0 0 | 8 8 8 8 0 0 0 0 0 0 0 0 | 4 4 4 4 0 0 0 0
.(o.). .(o.). .(o.). .(o.).    | * 4 * * ♦ 1 0 1 1  4 1 0 0 0 | 1 1  4 0 0 4 4 2 2 1 1  4 0 0 | 4 4 2 2 2 2 2 2 4 4 2 2 | 2 2 2 2 2 2 2 2
.(.o). .(.o). .(.o). .(.o).    | * * 4 * ♦ 0 1 0 0  4 0 1 1 1 | 0 0  4 1 1 2 2 4 4 0 0  4 1 1 | 2 2 4 4 2 2 2 2 2 2 4 4 | 2 2 2 2 2 2 2 2
.(..)o .(..)o .(..)o .(..)o    | * * * 1 ♦ 0 0 0 0  0 4 0 0 4 | 0 0  0 0 0 0 0 0 0 2 2 16 2 2 | 0 0 0 0 0 0 0 0 8 8 8 8 | 0 0 0 0 4 4 4 4
-------------------------------+---------+--------------------+-------------------------------+-------------------------+----------------
o(o.). o(o.). o(o.). o(o.).&#x | 1 1 0 0 | 4 * * *  * * * * * ♦ 1 1  4 0 0 0 0 0 0 0 0  0 0 0 | 4 4 2 2 0 0 0 0 0 0 0 0 | 2 2 2 2 0 0 0 0
o(.o). o(.o). o(.o). o(.o).&#x | 1 0 1 0 | * 4 * *  * * * * * ♦ 0 0  4 1 1 0 0 0 0 0 0  0 0 0 | 2 2 4 4 0 0 0 0 0 0 0 0 | 2 2 2 2 0 0 0 0
.(x.). .(..). .(..). .(..).    | 0 2 0 0 | * * 2 *  * * * * * ♦ 1 0  0 0 0 4 0 0 0 1 0  0 0 0 | 4 0 0 0 2 2 0 0 4 0 0 0 | 2 2 0 0 2 2 0 0
.(..). .(x.). .(..). .(..).    | 0 2 0 0 | * * * 2  * * * * * ♦ 0 1  0 0 0 0 4 0 0 0 1  0 0 0 | 0 4 0 0 0 0 0 2 2 0 4 0 | 0 0 2 2 0 0 2 2
.(oo). .(oo). .(oo). .(oo).&#x | 0 1 1 0 | * * * * 16 * * * * ♦ 0 0  1 0 0 1 1 1 1 0 0  1 0 0 | 1 1 1 1 1 1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 1
.(o.)o .(o.)o .(o.)o .(o.)o&#x | 0 1 0 1 | * * * *  * 4 * * * ♦ 0 0  0 0 0 0 0 0 0 1 1  4 0 0 | 0 0 0 0 0 0 0 0 4 4 2 2 | 0 0 0 0 2 2 2 2
.(..). .(..). .(.x). .(..).    | 0 0 2 0 | * * * *  * * 2 * * ♦ 0 0  0 1 0 0 0 4 0 0 0  0 1 0 | 0 0 4 0 2 0 2 0 0 0 4 0 | 2 0 2 0 2 0 2 0
.(..). .(..). .(..). .(.x).    | 0 0 2 0 | * * * *  * * * 2 * ♦ 0 0  0 0 1 0 0 0 4 0 0  0 0 1 | 0 0 0 4 0 2 0 2 0 0 0 4 | 0 2 0 2 0 2 0 2
.(.o)o .(.o)o .(.o)o .(.o)o&#x | 0 0 1 1 | * * * *  * * * * 4 ♦ 0 0  0 0 0 0 0 0 0 0 0  4 1 1 | 0 0 0 0 0 0 0 0 2 2 4 4 | 0 0 0 0 2 2 2 2
-------------------------------+---------+--------------------+-------------------------------+-------------------------+----------------
o(x.). .(..). .(..). .(..).&#x | 1 2 0 0 | 2 0 1 0  0 0 0 0 0 | 2 *  * * * * * * * * *  * * * | 4 0 0 0 0 0 0 0 0 0 0 0 | 2 2 0 0 0 0 0 0
.(..). o(x.). .(..). .(..).&#x | 1 2 0 0 | 2 0 0 1  0 0 0 0 0 | * 2  * * * * * * * * *  * * * | 0 4 0 0 0 0 0 0 0 0 0 0 | 0 0 2 2 0 0 0 0
o(oo). o(oo). o(oo). o(oo).&#x | 1 1 1 0 | 1 1 0 0  1 0 0 0 0 | * * 16 * * * * * * * *  * * * | 1 1 1 1 0 0 0 0 0 0 0 0 | 1 1 1 1 0 0 0 0
.(..). .(..). o(.x). .(..).&#x | 1 0 2 0 | 0 2 0 0  0 0 1 0 0 | * *  * 2 * * * * * * *  * * * | 0 0 4 0 0 0 0 0 0 0 0 0 | 2 0 2 0 0 0 0 0
.(..). .(..). .(..). o(.x).&#x | 1 0 2 0 | 0 2 0 0  0 0 0 1 0 | * *  * * 2 * * * * * *  * * * | 0 0 0 4 0 0 0 0 0 0 0 0 | 0 2 0 2 0 0 0 0
.(xo). .(..). .(..). .(..).&#x | 0 2 1 0 | 0 0 1 0  2 0 0 0 0 | * *  * * * 8 * * * * *  * * * | 1 0 0 0 1 1 0 0 1 0 0 0 | 1 1 0 0 1 1 0 0
.(..). .(xo). .(..). .(..).&#x | 0 2 1 0 | 0 0 0 1  2 0 0 0 0 | * *  * * * * 8 * * * *  * * * | 0 1 0 0 0 0 1 1 0 1 0 0 | 0 0 1 1 0 0 1 1
.(..). .(..). .(ox). .(..).&#x | 0 1 2 0 | 0 0 0 0  2 0 1 0 0 | * *  * * * * * 8 * * *  * * * | 0 0 1 0 1 0 1 0 0 0 1 0 | 1 0 1 0 1 0 1 0
.(..). .(..). .(..). .(ox).&#x | 0 1 2 0 | 0 0 0 0  2 0 0 1 0 | * *  * * * * * * 8 * *  * * * | 0 0 0 1 0 1 0 1 0 0 0 1 | 0 1 0 1 0 1 0 1
.(x.)o .(..). .(..). .(..).&#x | 0 2 0 1 | 0 0 1 0  0 2 0 0 0 | * *  * * * * * * * 2 *  * * * | 0 0 0 0 0 0 0 0 4 0 0 0 | 0 0 0 0 2 2 0 0
.(..). .(x.)o .(..). .(..).&#x | 0 2 0 1 | 0 0 0 1  0 2 0 0 0 | * *  * * * * * * * * 2  * * * | 0 0 0 0 0 0 0 0 0 4 0 0 | 0 0 0 0 0 0 2 2
.(oo)o .(oo)o .(oo)o .(oo)o&#x | 0 1 1 1 | 0 0 0 0  1 1 0 0 1 | * *  * * * * * * * * * 16 * * | 0 0 0 0 0 0 0 0 1 1 1 1 | 0 0 0 0 1 1 1 1
.(..). .(..). .(.x)o .(..).&#x | 0 0 2 1 | 0 0 0 0  0 0 1 0 2 | * *  * * * * * * * * *  * 2 * | 0 0 0 0 0 0 0 0 0 0 4 0 | 0 0 0 0 2 0 2 0
.(..). .(..). .(..). .(.x)o&#x | 0 0 2 1 | 0 0 0 0  0 0 0 1 2 | * *  * * * * * * * * *  * * 2 | 0 0 0 0 0 0 0 0 0 0 0 4 | 0 0 0 0 0 2 0 2
-------------------------------+---------+--------------------+-------------------------------+-------------------------+----------------
o(xo). .(..). .(..). .(..).&#x ♦ 1 2 1 0 | 2 1 1 0  2 0 0 0 0 | 1 0  2 0 0 1 0 0 0 0 0  0 0 0 | 8 * * * * * * * * * * * | 1 1 0 0 0 0 0 0
.(..). o(xo). .(..). .(..).&#x ♦ 1 2 1 0 | 2 1 0 1  2 0 0 0 0 | 0 1  2 0 0 0 1 0 0 0 0  0 0 0 | * 8 * * * * * * * * * * | 0 0 1 1 0 0 0 0
.(..). .(..). o(ox). .(..).&#x ♦ 1 1 2 0 | 1 2 0 0  2 0 1 0 0 | 0 0  2 1 0 0 0 1 0 0 0  0 0 0 | * * 8 * * * * * * * * * | 1 0 1 0 0 0 0 0
.(..). .(..). .(..). o(ox).&#x ♦ 1 1 2 0 | 1 2 0 0  2 0 0 1 0 | 0 0  2 0 1 0 0 0 1 0 0  0 0 0 | * * * 8 * * * * * * * * | 0 1 0 1 0 0 0 0
.(xo). .(..). .(ox). .(..).&#x ♦ 0 2 2 0 | 0 0 1 0  4 0 1 0 0 | 0 0  0 0 0 2 0 2 0 0 0  0 0 0 | * * * * 4 * * * * * * * | 1 0 0 0 1 0 0 0
.(xo). .(..). .(..). .(ox).&#x ♦ 0 2 2 0 | 0 0 1 0  4 0 0 1 0 | 0 0  0 0 0 2 0 0 2 0 0  0 0 0 | * * * * * 4 * * * * * * | 0 1 0 0 0 1 0 0
.(..). .(xo). .(ox). .(..).&#x ♦ 0 2 2 0 | 0 0 0 1  4 0 1 0 0 | 0 0  0 0 0 0 2 2 0 0 0  0 0 0 | * * * * * * 4 * * * * * | 0 0 1 0 0 0 1 0
.(..). .(xo). .(..). .(ox).&#x ♦ 0 2 2 0 | 0 0 0 1  4 0 0 1 0 | 0 0  0 0 0 0 2 0 2 0 0  0 0 0 | * * * * * * * 4 * * * * | 0 0 0 1 0 0 0 1
.(xo)o .(..). .(..). .(..).&#x ♦ 0 2 1 1 | 0 0 1 0  2 2 0 0 1 | 0 0  0 0 0 1 0 0 0 1 0  2 0 0 | * * * * * * * * 8 * * * | 0 0 0 0 1 1 0 0
.(..). .(xo)o .(..). .(..).&#x ♦ 0 2 1 1 | 0 0 0 1  2 2 0 0 1 | 0 0  0 0 0 0 1 0 0 0 1  2 0 0 | * * * * * * * * * 8 * * | 0 0 0 0 0 0 1 1
.(..). .(..). .(ox)o .(..).&#x ♦ 0 1 2 1 | 0 0 0 0  2 1 1 0 2 | 0 0  0 0 0 0 0 1 0 0 0  2 1 0 | * * * * * * * * * * 8 * | 0 0 0 0 1 0 1 0
.(..). .(..). .(..). .(ox)o&#x ♦ 0 1 2 1 | 0 0 0 0  2 1 0 1 2 | 0 0  0 0 0 0 0 0 1 0 0  2 0 1 | * * * * * * * * * * * 8 | 0 0 0 0 0 1 0 1
-------------------------------+---------+--------------------+-------------------------------+-------------------------+----------------
o(xo). .(..). o(ox). .(..).&#x ♦ 1 2 2 0 | 2 2 1 0  4 0 1 0 0 | 1 0  4 1 0 2 0 2 0 0 0  0 0 0 | 2 0 2 0 1 0 0 0 0 0 0 0 | 4 * * * * * * *
o(xo). .(..). .(..). o(ox).&#x ♦ 1 2 2 0 | 2 2 1 0  4 0 0 1 0 | 1 0  4 0 1 2 0 0 2 0 0  0 0 0 | 2 0 0 2 0 1 0 0 0 0 0 0 | * 4 * * * * * *
.(..). o(xo). o(ox). .(..).&#x ♦ 1 2 2 0 | 2 2 0 1  4 0 1 0 0 | 0 1  4 1 0 0 2 2 0 0 0  0 0 0 | 0 2 2 0 0 0 1 0 0 0 0 0 | * * 4 * * * * *
.(..). o(xo). .(..). o(ox).&#x ♦ 1 2 2 0 | 2 2 0 1  4 0 0 1 0 | 0 1  4 0 1 0 2 0 2 0 0  0 0 0 | 0 2 0 2 0 0 0 1 0 0 0 0 | * * * 4 * * * *
.(xo)o .(..). .(ox)o .(..).&#x ♦ 0 2 2 1 | 0 0 1 0  4 2 1 0 2 | 0 0  0 0 0 2 0 2 0 1 0  4 1 0 | 0 0 0 0 1 0 0 0 2 0 2 0 | * * * * 4 * * *
.(xo)o .(..). .(..). .(ox)o&#x ♦ 0 2 2 1 | 0 0 1 0  4 2 0 1 2 | 0 0  0 0 0 2 0 0 2 1 0  4 0 1 | 0 0 0 0 0 1 0 0 2 0 0 2 | * * * * * 4 * *
.(..). .(xo)o .(ox)o .(..).&#x ♦ 0 2 2 1 | 0 0 0 1  4 2 1 0 2 | 0 0  0 0 0 0 2 2 0 0 1  4 1 0 | 0 0 0 0 0 0 1 0 0 2 2 0 | * * * * * * 4 *
.(..). .(xo)o .(..). .(ox)o&#x ♦ 0 2 2 1 | 0 0 0 1  4 2 0 1 2 | 0 0  0 0 0 0 2 0 2 0 1  4 0 1 | 0 0 0 0 0 0 0 1 0 2 0 2 | * * * * * * * 4

top of page