Acronym	rap (old: rip, alt.: tetaoct), tet || oct, K-4.5
Name	rectified pentachoron, pyrorectichoron, tetrahedral cupola, vertex figure of hin, tetrahedron atop octahedron, Gosset polytope 02,1
Segmentochoron display / VRML	⭳
Cross sections	©
Circumradius	sqrt(3/5) = 0.774597
Inradius wrt. tet	3/sqrt(40) = 0.474342
Inradius wrt. oct	1/sqrt(10) = 0.316228
Vertex figure	©
Vertex layers	Layer Symmetry Subsymmetries   o3o3o3o o3o3o . o3o . o o . o3o . o3o3o 1 o3x3o3o o3x3o . oct first o3x . o {3} first o . o3o vertex first . x3o3o tet first 2 x3o3o . opposite tet x3o . x x . x3o vertex figure . o3x3o opposite oct 3   o3o . o opposite vertex o . o3x opposite {3}
Lace city in approx. ASCII-art	o3o x3o -- tet x3o o3x -- oct
	x o o x -- tet o o x x o o -- oct \ \ +-- {3} \ +------- gyro trip +------------ point
Lace hyper city in approx. ASCII-art	©   o o x o layer of tet o o o x layer of tet o o o layer of oct o o o o x x layer of oct o o o layer of oct
	©   o point x x x trip o o o dual {3}
Volume	11 sqrt(5)/96 = 0.256216
Surface	25 sqrt(2)/12 = 2.946278
General of army	(is itself convex)
Colonel of regiment	(is itself locally convex – uniform polychoral members: by cells: oct tet trip rap 5 5 0 pinnip 5 0 10 firp 0 5 10 )
Dual	o3m3o3o
Dihedral angles	at {3} between oct and tet:   arccos(-1/4) = 104.477512° at {3} between oct and oct:   arccos(1/4) = 75.522488°
Face vector	10, 30, 30, 10
Confer	Grünbaumian relatives: firp+rap+15{4}   2rap   2rap+20tet   related segmentochora: trippy   bidrap   traf   pafirp   fitetaoct   hotetahoct   variations: qo3oq3oo&#x   blends: turap   aurap   ambification: srip   ambification pre-image: pen   general polytopal classes: Wythoffian polychora   segmentochora   fundamental lace prisms   bistratic lace towers   lace simplices   Coxeter-Elte-Gosset polytopes   analogs: rectified simplex rSn   birectified simplex brSn   Gossetic n2,1   rectified Gossetic r(n2,1)   rectified Gossetic r(1n,2)
External links

 ©    ©

A 4D representation of the Petersen graph can be obtained by the vertices and the diagonals of the octs of this polychoron, cf. the left picture on the right. Even though, the fact that 3 lines then do cross perpendicularily each, happens to be a matter of this representation only. Geometrically, the Petersen graph also is the graph formed by the vertices and edges of the hemi-dodecahedron (eldoe), that is, a dodecahedron where opposite points, lines and faces got identified.

The 10 vertices, and 10 of the triangle faces represent a self dual symmetric Desargues configuration, (10₃), cf. the rightmost picture. It has the full 120 automorphisms of the rectified 5-cell.

Incidence matrix according to Dynkin symbol

o3x3o3o

. . . . | 10 ♦  6 |  3  6 | 3 2
--------+----+----+-------+----
. x . . |  2 | 30 |  1  2 | 2 1
--------+----+----+-------+----
o3x . . |  3 |  3 | 10  * | 2 0
. x3o . |  3 |  3 |  * 20 | 1 1
--------+----+----+-------+----
o3x3o . ♦  6 | 12 |  4  4 | 5 *
. x3o3o ♦  4 |  6 |  0  4 | * 5

snubbed forms: o3β3o3o

o3x3o3/2o

. . .   . | 10 ♦  6 |  3  6 | 3 2
----------+----+----+-------+----
. x .   . |  2 | 30 |  1  2 | 2 1
----------+----+----+-------+----
o3x .   . |  3 |  3 | 10  * | 2 0
. x3o   . |  3 |  3 |  * 20 | 1 1
----------+----+----+-------+----
o3x3o   . ♦  6 | 12 |  4  4 | 5 *
. x3o3/2o ♦  4 |  6 |  0  4 | * 5

o3x3/2o3o

. .   . . | 10 ♦  6 |  3  6 | 3 2
----------+----+----+-------+----
. x   . . |  2 | 30 |  1  2 | 2 1
----------+----+----+-------+----
o3x   . . |  3 |  3 | 10  * | 2 0
. x3/2o . |  3 |  3 |  * 20 | 1 1
----------+----+----+-------+----
o3x3/2o . ♦  6 | 12 |  4  4 | 5 *
. x3/2o3o ♦  4 |  6 |  0  4 | * 5

o3x3/2o3/2o

. .   .   . | 10 ♦  6 |  3  6 | 3 2
------------+----+----+-------+----
. x   .   . |  2 | 30 |  1  2 | 2 1
------------+----+----+-------+----
o3x   .   . |  3 |  3 | 10  * | 2 0
. x3/2o   . |  3 |  3 |  * 20 | 1 1
------------+----+----+-------+----
o3x3/2o   . ♦  6 | 12 |  4  4 | 5 *
. x3/2o3/2o ♦  4 |  6 |  0  4 | * 5

o3/2x3o3o

.   . . . | 10 ♦  6 |  3  6 | 3 2
----------+----+----+-------+----
.   x . . |  2 | 30 |  1  2 | 2 1
----------+----+----+-------+----
o3/2x . . |  3 |  3 | 10  * | 2 0
.   x3o . |  3 |  3 |  * 20 | 1 1
----------+----+----+-------+----
o3/2x3o . ♦  6 | 12 |  4  4 | 5 *
.   x3o3o ♦  4 |  6 |  0  4 | * 5

o3/2x3o3/2o

.   . .   . | 10 ♦  6 |  3  6 | 3 2
------------+----+----+-------+----
.   x .   . |  2 | 30 |  1  2 | 2 1
------------+----+----+-------+----
o3/2x .   . |  3 |  3 | 10  * | 2 0
.   x3o   . |  3 |  3 |  * 20 | 1 1
------------+----+----+-------+----
o3/2x3o   . ♦  6 | 12 |  4  4 | 5 *
.   x3o3/2o ♦  4 |  6 |  0  4 | * 5

o3/2x3/2o3o

.   .   . . | 10 ♦  6 |  3  6 | 3 2
------------+----+----+-------+----
.   x   . . |  2 | 30 |  1  2 | 2 1
------------+----+----+-------+----
o3/2x   . . |  3 |  3 | 10  * | 2 0
.   x3/2o . |  3 |  3 |  * 20 | 1 1
------------+----+----+-------+----
o3/2x3/2o . ♦  6 | 12 |  4  4 | 5 *
.   x3/2o3o ♦  4 |  6 |  0  4 | * 5

o3/2x3/2o3/2o

.   .   .   . | 10 ♦  6 |  3  6 | 3 2
--------------+----+----+-------+----
.   x   .   . |  2 | 30 |  1  2 | 2 1
--------------+----+----+-------+----
o3/2x   .   . |  3 |  3 | 10  * | 2 0
.   x3/2o   . |  3 |  3 |  * 20 | 1 1
--------------+----+----+-------+----
o3/2x3/2o   . ♦  6 | 12 |  4  4 | 5 *
.   x3/2o3/2o ♦  4 |  6 |  0  4 | * 5

xo3ox3oo&#x   → height = sqrt(5/8) = 0.790569
(tet || oct)

o.3o.3o.    | 4 * ♦ 3  3  0 | 3 3  3 0 0 | 1 3 1 0
.o3.o3.o    | * 6 ♦ 0  2  4 | 0 1  4 2 2 | 0 2 2 1
------------+-----+---------+------------+--------
x. .. ..    | 2 0 | 6  *  * | 2 1  0 0 0 | 1 2 0 0
oo3oo3oo&#x | 1 1 | * 12  * | 0 1  2 0 0 | 0 2 1 0
.. .x ..    | 0 2 | *  * 12 | 0 0  1 1 1 | 0 1 1 1
------------+-----+---------+------------+--------
x.3o. ..    | 3 0 | 3  0  0 | 4 *  * * * | 1 1 0 0
xo .. ..&#x | 2 1 | 1  2  0 | * 6  * * * | 0 2 0 0
.. ox ..&#x | 1 2 | 0  2  1 | * * 12 * * | 0 1 1 0
.o3.x ..    | 0 3 | 0  0  3 | * *  * 4 * | 0 1 0 1
.. .x3.o    | 0 3 | 0  0  3 | * *  * * 4 | 0 0 1 1
------------+-----+---------+------------+--------
x.3o.3o.    ♦ 4 0 | 6  0  0 | 4 0  0 0 0 | 1 * * *
xo3ox ..&#x ♦ 3 3 | 3  6  3 | 1 3  3 1 0 | * 4 * *
.. ox3oo&#x ♦ 1 3 | 0  3  3 | 0 0  3 0 1 | * * 4 *
.o3.x3.o    ♦ 0 6 | 0  0 12 | 0 0  0 4 4 | * * * 1

oxo oxo3oox&#xt   → both heights = sqrt(5/12) = 0.645497
(pt || pseudo trip || dual {3})

o.. o..3o..     | 1 * * ♦ 6 0 0  0 0 | 3 6 0 0 0 0 0 | 3 2 0 0
.o. .o.3.o.     | * 6 * ♦ 1 1 2  2 0 | 1 2 1 2 2 1 0 | 2 1 1 1
..o ..o3..o     | * * 3 ♦ 0 0 0  4 2 | 0 0 0 2 2 4 1 | 1 0 2 2
----------------+-------+------------+---------------+--------
oo. oo.3oo.&#x  | 1 1 0 | 6 * *  * * | 1 2 0 0 0 0 0 | 2 1 0 0
.x. ... ...     | 0 2 0 | * 3 *  * * | 1 0 0 2 0 0 0 | 2 0 1 0
... .x. ...     | 0 2 0 | * * 6  * * | 0 1 1 0 1 0 0 | 1 1 0 1
.oo .oo3.oo&#x  | 0 1 1 | * * * 12 * | 0 0 0 1 1 1 0 | 1 0 1 1
... ... ..x     | 0 0 2 | * * *  * 3 | 0 0 0 0 0 2 1 | 0 0 1 2
----------------+-------+------------+---------------+--------
ox. ... ...&#x  | 1 2 0 | 2 1 0  0 0 | 3 * * * * * * | 2 0 0 0
... ox. ...&#x  | 1 2 0 | 2 0 1  0 0 | * 6 * * * * * | 1 1 0 0
... .x.3.o.     | 0 3 0 | 0 0 3  0 0 | * * 2 * * * * | 0 1 0 1
.xo ... ...&#x  | 0 2 1 | 0 1 0  2 0 | * * * 6 * * * | 1 0 1 0
... .xo ...&#x  | 0 2 1 | 0 0 1  2 0 | * * * * 6 * * | 1 0 0 1
... ... .ox&#x  | 0 1 2 | 0 0 0  2 1 | * * * * * 6 * | 0 0 1 1
... ..o3..x     | 0 0 3 | 0 0 0  0 3 | * * * * * * 1 | 0 0 0 2
----------------+-------+------------+---------------+--------
oxo oxo ...&#xt ♦ 1 4 1 | 4 2 2  4 0 | 2 2 0 2 2 0 0 | 3 * * *
... ox.3oo.&#x  ♦ 1 3 0 | 3 0 3  0 0 | 0 3 1 0 0 0 0 | * 2 * *
.xo ... .ox&#x  ♦ 0 2 2 | 0 1 0  4 1 | 0 0 0 2 0 2 0 | * * 3 *
... .xo3.ox&#x  ♦ 0 3 3 | 0 0 3  6 3 | 0 0 1 0 3 3 1 | * * * 2

o(xx)(oo) o(xo)(ox)&#xt   → both heights = sqrt(5/12) = 0.645497
(pt || pseudo trip || dual {3})

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

oxoo3xoxo&#xr   → all heights = sqrt(2/3) = 0.816497
({3} || pseudo (dual {3} || pt) || {3})

o...3o...     | 3 * * * ♦ 2 2 1 1 0 0 0 0 | 1 1 2 2 2 1 0 0 0 0 0 | 1 1 1 2 0 0
.o..3.o..     | * 3 * * ♦ 0 2 0 0 2 2 0 0 | 0 2 1 0 2 0 1 2 1 0 0 | 1 0 2 1 1 0
..o.3..o.     | * * 3 * ♦ 0 0 1 0 0 2 2 1 | 0 0 0 0 2 1 0 1 2 1 2 | 0 0 1 2 1 1
...o3...o     | * * * 1 ♦ 0 0 0 3 0 0 0 3 | 0 0 0 3 0 3 0 0 0 0 3 | 0 1 0 3 0 1
--------------+---------+-----------------+-----------------------+------------
.... x...     | 2 0 0 0 | 3 * * * * * * * | 1 0 1 1 0 0 0 0 0 0 0 | 1 1 0 1 0 0
oo..3oo..&#x  | 1 1 0 0 | * 6 * * * * * * | 0 1 1 0 1 0 0 0 0 0 0 | 1 0 1 1 0 0
o.o.3o.o.&#x  | 1 0 1 0 | * * 3 * * * * * | 0 0 0 0 2 1 0 0 0 0 0 | 0 0 1 2 0 0
o..o3o..o&#x  | 1 0 0 1 | * * * 3 * * * * | 0 0 0 2 0 1 0 0 0 0 0 | 0 1 0 2 0 0
.x.. ....     | 0 2 0 0 | * * * * 3 * * * | 0 1 0 0 0 0 1 1 0 0 0 | 1 0 1 0 1 0
.oo.3.oo.&#x  | 0 1 1 0 | * * * * * 6 * * | 0 0 0 0 1 0 0 1 1 0 0 | 0 0 1 1 1 0
.... ..x.     | 0 0 2 0 | * * * * * * 3 * | 0 0 0 0 0 0 0 0 1 1 1 | 0 0 0 1 1 1
..oo3..oo&#x  | 0 0 1 1 | * * * * * * * 3 | 0 0 0 0 0 1 0 0 0 0 2 | 0 0 0 2 0 1
--------------+---------+-----------------+-----------------------+------------
o...3x...     | 3 0 0 0 | 3 0 0 0 0 0 0 0 | 1 * * * * * * * * * * | 1 1 0 0 0 0
ox.. ....&#x  | 1 2 0 0 | 0 2 0 0 1 0 0 0 | * 3 * * * * * * * * * | 1 0 1 0 0 0
.... xo..&#x  | 2 1 0 0 | 1 2 0 0 0 0 0 0 | * * 3 * * * * * * * * | 1 0 0 1 0 0
.... x..o&#x  | 2 0 0 1 | 1 0 0 2 0 0 0 0 | * * * 3 * * * * * * * | 0 1 0 1 0 0
ooo.3ooo.&#x  | 1 1 1 0 | 0 1 1 0 0 1 0 0 | * * * * 6 * * * * * * | 0 0 1 1 0 0
o.oo3o.oo&#x  | 1 0 1 1 | 0 0 1 1 0 0 0 1 | * * * * * 3 * * * * * | 0 0 0 2 0 0
.x..3.o..     | 0 3 0 0 | 0 0 0 0 3 0 0 0 | * * * * * * 1 * * * * | 1 0 0 0 1 0
.xo. ....&#x  | 0 2 1 0 | 0 0 0 0 1 2 0 0 | * * * * * * * 3 * * * | 0 0 1 0 1 0
.... .ox.&#x  | 0 1 2 0 | 0 0 0 0 0 2 1 0 | * * * * * * * * 3 * * | 0 0 0 1 1 0
..o.3..x.     | 0 0 3 0 | 0 0 0 0 0 0 3 0 | * * * * * * * * * 1 * | 0 0 0 0 1 1
.... ..xo&#x  | 0 0 2 1 | 0 0 0 0 0 0 1 2 | * * * * * * * * * * 3 | 0 0 0 1 0 1
--------------+---------+-----------------+-----------------------+------------
ox..3xo..&#x  ♦ 3 3 0 0 | 3 6 0 0 3 0 0 0 | 1 3 3 0 0 0 1 0 0 0 0 | 1 * * * * *
o..o3x..o&#x  ♦ 3 0 0 1 | 3 0 0 3 0 0 0 0 | 1 0 0 3 0 0 0 0 0 0 0 | * 1 * * * *
oxo. ....&#x  ♦ 1 2 1 0 | 0 2 1 0 1 2 0 0 | 0 1 0 0 2 0 0 1 0 0 0 | * * 3 * * *
.... xoxo&#xr ♦ 2 1 2 1 | 1 2 2 2 0 2 1 2 | 0 0 1 1 2 2 0 0 1 0 1 | * * * 3 * *
.xo.3.ox.&#x  ♦ 0 3 3 0 | 0 0 0 0 3 6 3 0 | 0 0 0 0 0 0 1 3 3 1 0 | * * * * 1 *
..oo3..xo&#x  ♦ 0 0 3 1 | 0 0 0 0 0 0 3 3 | 0 0 0 0 0 0 0 0 0 1 3 | * * * * * 1

oxoox oxoxo&#xr   → height(1,2) = height(2,3) = height(4,5) = 1/sqrt(2) = 0.707107
                    height(1,5) = height(3,4) = sqrt(3)/2 = 0.866025
(layer 1-3: oct-base, layer 4+5: tet-base)

o.... o....     | 1 * * * * ♦ 4 2 0 0 0 0 0 0 0 0 0 | 2 2 1 4 0 0 0 0 0 0 0 0 0 | 1 2 2 0 0 0
.o... .o...     | * 4 * * * ♦ 1 0 1 1 1 1 1 0 0 0 0 | 1 1 0 1 1 1 1 1 1 1 0 0 0 | 1 1 1 1 1 0
..o.. ..o..     | * * 1 * * ♦ 0 0 0 0 4 0 0 2 0 0 0 | 0 0 0 0 2 2 0 0 4 0 1 0 0 | 1 0 0 2 2 0
...o. ...o.     | * * * 2 * ♦ 0 0 0 0 0 2 0 1 1 2 0 | 0 0 0 0 0 0 1 0 2 2 1 1 2 | 0 1 0 1 2 1
....o ....o     | * * * * 2 ♦ 0 1 0 0 0 0 2 0 0 2 1 | 0 0 1 2 0 0 0 1 0 2 0 2 1 | 0 2 1 0 1 1
----------------+-----------+-----------------------+---------------------------+------------
oo... oo...&#x  | 1 1 0 0 0 | 4 * * * * * * * * * * | 1 1 0 1 0 0 0 0 0 0 0 0 0 | 1 1 1 0 0 0
o...o o...o&#x  | 1 0 0 0 1 | * 2 * * * * * * * * * | 0 0 1 2 0 0 0 0 0 0 0 0 0 | 0 2 1 0 0 0
.x... .....     | 0 2 0 0 0 | * * 2 * * * * * * * * | 1 0 0 0 1 0 1 0 0 0 0 0 0 | 1 1 0 1 0 0
..... .x...     | 0 2 0 0 0 | * * * 2 * * * * * * * | 0 1 0 0 0 1 0 1 0 0 0 0 0 | 1 0 1 0 1 0
.oo.. .oo..&#x  | 0 1 1 0 0 | * * * * 4 * * * * * * | 0 0 0 0 1 1 0 0 1 0 0 0 0 | 1 0 0 1 1 0
.o.o. .o.o.&#x  | 0 1 0 1 0 | * * * * * 4 * * * * * | 0 0 0 0 0 0 1 0 1 1 0 0 0 | 0 1 0 1 1 0
.o..o .o..o&#x  | 0 1 0 0 1 | * * * * * * 4 * * * * | 0 0 0 1 0 0 0 1 0 1 0 0 0 | 0 1 1 0 1 0
..oo. ..oo.&#x  | 0 0 1 1 0 | * * * * * * * 2 * * * | 0 0 0 0 0 0 0 0 2 0 1 0 0 | 0 0 0 1 2 0
..... ...x.     | 0 0 0 2 0 | * * * * * * * * 1 * * | 0 0 0 0 0 0 0 0 0 0 1 0 2 | 0 0 0 0 2 1
...oo ...oo&#x  | 0 0 0 1 1 | * * * * * * * * * 4 * | 0 0 0 0 0 0 0 0 0 1 0 1 1 | 0 1 0 0 1 1
....x .....     | 0 0 0 0 2 | * * * * * * * * * * 1 | 0 0 1 0 0 0 0 0 0 0 0 2 0 | 0 2 0 0 0 1
----------------+-----------+-----------------------+---------------------------+------------
ox... .....&#x  | 1 2 0 0 0 | 2 0 1 0 0 0 0 0 0 0 0 | 2 * * * * * * * * * * * * | 1 1 0 0 0 0
..... ox...&#x  | 1 2 0 0 0 | 2 0 0 1 0 0 0 0 0 0 0 | * 2 * * * * * * * * * * * | 1 0 1 0 0 0
o...x .....&#x  | 1 0 0 0 2 | 0 2 0 0 0 0 0 0 0 0 1 | * * 1 * * * * * * * * * * | 0 2 0 0 0 0
oo..o oo..o&#x  | 1 1 0 0 1 | 1 1 0 0 0 0 1 0 0 0 0 | * * * 4 * * * * * * * * * | 0 1 1 0 0 0
.xo.. .....&#x  | 0 2 1 0 0 | 0 0 1 0 2 0 0 0 0 0 0 | * * * * 2 * * * * * * * * | 1 0 0 1 0 0
..... .xo..&#x  | 0 2 1 0 0 | 0 0 0 1 2 0 0 0 0 0 0 | * * * * * 2 * * * * * * * | 1 0 0 0 1 0
.x.o. .....&#x  | 0 2 0 1 0 | 0 0 1 0 0 2 0 0 0 0 0 | * * * * * * 2 * * * * * * | 0 1 0 1 0 0
..... .x..o&#x  | 0 2 0 0 1 | 0 0 0 1 0 0 2 0 0 0 0 | * * * * * * * 2 * * * * * | 0 0 1 0 1 0
.ooo. .ooo.&#x  | 0 1 1 1 0 | 0 0 0 0 1 1 0 1 0 0 0 | * * * * * * * * 4 * * * * | 0 0 0 1 1 0
.o.oo .o.oo&#x  | 0 1 0 1 1 | 0 0 0 0 0 1 1 0 0 1 0 | * * * * * * * * * 4 * * * | 0 1 0 0 1 0
..... ..ox.&#x  | 0 0 1 2 0 | 0 0 0 0 0 0 0 2 1 0 0 | * * * * * * * * * * 1 * * | 0 0 0 0 2 0
...ox .....&#x  | 0 0 0 1 2 | 0 0 0 0 0 0 0 0 0 2 1 | * * * * * * * * * * * 2 * | 0 1 0 0 0 1
..... ...xo&#x  | 0 0 0 2 1 | 0 0 0 0 0 0 0 0 1 2 0 | * * * * * * * * * * * * 2 | 0 0 0 0 1 1
----------------+-----------+-----------------------+---------------------------+------------
oxo.. oxo..&#xt ♦ 1 4 1 0 0 | 4 0 2 2 4 0 0 0 0 0 0 | 2 2 0 0 2 2 0 0 0 0 0 0 0 | 1 * * * * *
ox.ox .....&#xr ♦ 1 2 0 1 2 | 2 2 1 0 0 2 2 0 0 2 1 | 1 0 1 2 0 0 1 0 0 2 0 1 0 | * 2 * * * *
..... ox..o&#x  ♦ 1 2 0 0 1 | 2 1 0 1 0 0 2 0 0 0 0 | 0 1 0 2 0 0 0 1 0 0 0 0 0 | * * 2 * * *
.xoo. .....&#x  ♦ 0 2 1 1 0 | 0 0 1 0 2 2 0 1 0 0 0 | 0 0 0 0 1 0 1 0 2 0 0 0 0 | * * * 2 * *
..... .xoxo&#xr ♦ 0 2 1 2 1 | 0 0 0 1 2 2 2 2 1 2 0 | 0 0 0 0 0 1 0 1 2 2 1 0 1 | * * * * 2 *
...ox ...xo&#x  ♦ 0 0 0 2 2 | 0 0 0 0 0 0 0 0 1 4 1 | 0 0 0 0 0 0 0 0 0 0 0 2 2 | * * * * * 1

or
o.... o....     & | 2 * * ♦ 4 2 0 0 0 0 | 2 2 1 4 0 0 0 | 1 2 2 0  A=C
.o... .o...       | * 4 * ♦ 2 0 2 2 0 0 | 2 2 0 2 2 1 0 | 1 2 2 0  B
...o. ...o.     & | * * 4 ♦ 0 1 0 2 1 2 | 0 0 1 2 1 2 3 | 0 3 1 1  D=E
------------------+-------+-------------+---------------+--------
oo... oo...&#x  & | 1 1 0 | 8 * * * * * | 1 1 0 1 0 0 0 | 1 1 1 0
o...o o...o&#x  & | 1 0 1 | * 4 * * * * | 0 0 1 2 0 0 0 | 0 2 1 0
.x... .....     & | 0 2 0 | * * 4 * * * | 1 1 0 0 1 0 0 | 1 1 1 0
.o.o. .o.o.&#x  & | 0 1 1 | * * * 8 * * | 0 0 0 1 1 1 0 | 0 2 1 0
..... ...x.     & | 0 0 2 | * * * * 2 * | 0 0 1 0 0 0 2 | 0 2 0 1
...oo ...oo&#x    | 0 0 2 | * * * * * 4 | 0 0 0 0 0 1 2 | 0 2 0 1
------------------+-------+-------------+---------------+--------
ox... .....&#x  & | 1 2 0 | 2 0 1 0 0 0 | 4 * * * * * * | 1 1 0 0
..... ox...&#x  & | 1 2 0 | 2 0 1 0 0 0 | * 4 * * * * * | 1 0 1 0
o...x .....&#x  & | 1 0 2 | 0 2 0 0 1 0 | * * 2 * * * * | 0 2 0 0
oo..o oo..o&#x  & | 1 1 1 | 1 1 0 1 0 0 | * * * 8 * * * | 0 1 1 0
.x.o. .....&#x  & | 0 2 1 | 0 0 1 2 0 0 | * * * * 4 * * | 0 1 1 0
.o.oo .o.oo&#x    | 0 1 2 | 0 0 0 2 0 1 | * * * * * 4 * | 0 2 0 0
...ox .....&#x  & | 0 0 3 | 0 0 0 0 1 2 | * * * * * * 4 | 0 1 0 1
------------------+-------+-------------+---------------+--------
oxo.. oxo..&#xt   ♦ 2 4 0 | 8 0 4 0 0 0 | 4 4 0 0 0 0 0 | 1 * * *
ox.ox .....&#xr & ♦ 1 2 3 | 2 2 1 4 1 2 | 1 0 1 2 1 2 1 | * 4 * *
..... ox..o&#x  & ♦ 1 2 1 | 2 1 1 2 0 0 | 0 1 0 2 1 0 0 | * * 4 *
...ox ...xo&#x    ♦ 0 0 4 | 0 0 0 0 2 4 | 0 0 0 0 0 0 4 | * * * 1