Acronym rap (old: rip, alt.: tetaoct), tet || oct, K-4.5
Name rectified pentachoron,
tetrahedral cupola,
vertex figure of hin,
tetrahedron atop octahedron
 
Segmentochoron display
Cross sections
 ©
Circumradius sqrt(3/5) = 0.774597
Vertex figure
 ©
Vertex layers
LayerSymmetrySubsymmetries
 o3o3o3o o3o3o . o3o . o o . o3o . o3o3o
1o3x3o3o 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   
            
   x3o   o3x
   x o   o x   
               
o o   x x   o o
General of army (is itself convex)
Colonel of regiment (is itself locally convex – uniform polychoral members:
by cells: oct tet trip
rap 550
pinnip 5010
firp 0510
)
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°
Confer
Grünbaumian relatives:
firp+rap+15{4}   2rap   2rap+20tet  
related segmentochora:
trippy   traf   bidrap  
blends:
turap  
general polytopal classes:
segmentochora   fundamental lace prisms   bistratic lace towers  
External
links
hedrondude   wikipedia   WikiChoron   quickfur

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

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

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

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