rit

Acronym rit

Name rectified tesseract,
rectified octachoron,
birectified hexadecachoron

Cross sections

©

Circumradius sqrt(3/2) = 1.224745

Vertex figure

©

Vertex layers

Layer	Symmetry	Subsymmetries
	`o3o3o4o`	`o3o3o .`	`o3o . o`	`o . o4o`	`. o3o4o`
1	`o3o3x4o`	`o3o3x .` tet first	`o3o . o` vertex first	`o . x4o` {4} first	`. o3x4o` co first
2		`o3x3x .`	`o3x . q` vertex figure	`x . o4q`	`. o3o4q`
3		`x3x3o .`	`o3u . o`	`u . x4o`	`. o3x4o` opposite co
4		`x3o3o .` opposite tet	`x3x . q`	`x . o4q`
5			`u3o . o`	`o . x4o` opposite {4}
6			`x3o . q`
7			`o3o . o` opposite vertex
	`o3o3o *b3o`	`o3o3o .`	`o3o . *b3o`	`o . o o`	`. o3o *b3o`
1	`x3o3x *b3o`	`x3o3x .` co first	`x3o . *b3o` tet first	`x . x o` {4} first	`. o3x *b3o` tet first
2a		`o3o3u .`	`x3x . *b3o`	`q . o x`	`. x3x *b3o`
2b		`u3o3o .`	`x3x . *b3o`	`o . q x`	`. x3x *b3o`
3		`x3o3x .` opposite co	`o3x . *b3x`	`x . x u`	`. x3o *b3x`
4a			`o3o . *b3x` opposite tet	`q . o x`	`. o3o *b3x` opposite tet
4b			`o3o . *b3x` opposite tet	`o . q x`	`. o3o *b3x` opposite tet
5				`x . x o` opposite {4}

Lace city
in approx. ASCII-art

  x3o x3x o3x  
               
o3o u3o o3u o3o
               
  x3o x3x o3x

©	x4o o4q x4o o4q o4q x4o o4q x4o

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

General of army (is itself convex)

Colonel of regiment

(is itself locally convex – uniform polychoral members:

by cells:	co	oho	tet	tut
rit	8	0	16	0
hinnit	8	0	0	16
sto	0	8	8	8
firt	0	0	16	16

& others)

Confer

Grünbaumian relatives:: 2rit+64tet
segmentochora:: tet || tut tut || inv tut

External
links

Incidence matrix according to Dynkin symbol

o3o3x4o

. . . . | 32 ♦  6 |  6  3 |  2 3
--------+----+----+-------+-----
. . x . |  2 | 96 |  1  2 |  1 2
--------+----+----+-------+-----
. o3x . |  3 |  3 | 64  * |  1 1
. . x4o |  4 |  4 |  * 24 |  0 2
--------+----+----+-------+-----
o3o3x . ♦  4 |  6 |  4  0 | 16 *
. o3x4o ♦ 12 | 24 |  8  6 |  * 8

o3o3x4/3o

. . .   . | 32 ♦  6 |  6  3 |  2 3
----------+----+----+-------+-----
. . x   . |  2 | 96 |  1  2 |  1 2
----------+----+----+-------+-----
. o3x   . |  3 |  3 | 64  * |  1 1
. . x4/3o |  4 |  4 |  * 24 |  0 2
----------+----+----+-------+-----
o3o3x   . ♦  4 |  6 |  4  0 | 16 *
. o3x4/3o ♦ 12 | 24 |  8  6 |  * 8

o3o3/2x4o

. .   . . | 32 ♦  6 |  6  3 |  2 3
----------+----+----+-------+-----
. .   x . |  2 | 96 |  1  2 |  1 2
----------+----+----+-------+-----
. o3/2x . |  3 |  3 | 64  * |  1 1
. .   x4o |  4 |  4 |  * 24 |  0 2
----------+----+----+-------+-----
o3o3/2x . ♦  4 |  6 |  4  0 | 16 *
. o3/2x4o ♦ 12 | 24 |  8  6 |  * 8

o3o3/2x4/3o

. .   .   . | 32 ♦  6 |  6  3 |  2 3
------------+----+----+-------+-----
. .   x   . |  2 | 96 |  1  2 |  1 2
------------+----+----+-------+-----
. o3/2x   . |  3 |  3 | 64  * |  1 1
. .   x4/3o |  4 |  4 |  * 24 |  0 2
------------+----+----+-------+-----
o3o3/2x   . ♦  4 |  6 |  4  0 | 16 *
. o3/2x4/3o ♦ 12 | 24 |  8  6 |  * 8

o3/2o3x4o

.   . . . | 32 ♦  6 |  6  3 |  2 3
----------+----+----+-------+-----
.   . x . |  2 | 96 |  1  2 |  1 2
----------+----+----+-------+-----
.   o3x . |  3 |  3 | 64  * |  1 1
.   . x4o |  4 |  4 |  * 24 |  0 2
----------+----+----+-------+-----
o3/2o3x . ♦  4 |  6 |  4  0 | 16 *
.   o3x4o ♦ 12 | 24 |  8  6 |  * 8

o3/2o3x4/3o

.   . .   . | 32 ♦  6 |  6  3 |  2 3
------------+----+----+-------+-----
.   . x   . |  2 | 96 |  1  2 |  1 2
------------+----+----+-------+-----
.   o3x   . |  3 |  3 | 64  * |  1 1
.   . x4/3o |  4 |  4 |  * 24 |  0 2
------------+----+----+-------+-----
o3/2o3x   . ♦  4 |  6 |  4  0 | 16 *
.   o3x4/3o ♦ 12 | 24 |  8  6 |  * 8

o3/2o3/2x4o

.   .   . . | 32 ♦  6 |  6  3 |  2 3
------------+----+----+-------+-----
.   .   x . |  2 | 96 |  1  2 |  1 2
------------+----+----+-------+-----
.   o3/2x . |  3 |  3 | 64  * |  1 1
.   .   x4o |  4 |  4 |  * 24 |  0 2
------------+----+----+-------+-----
o3/2o3/2x . ♦  4 |  6 |  4  0 | 16 *
.   o3/2x4o ♦ 12 | 24 |  8  6 |  * 8

o3/2o3/2x4/3o

.   .   .   . | 32 ♦  6 |  6  3 |  2 3
--------------+----+----+-------+-----
.   .   x   . |  2 | 96 |  1  2 |  1 2
--------------+----+----+-------+-----
.   o3/2x   . |  3 |  3 | 64  * |  1 1
.   .   x4/3o |  4 |  4 |  * 24 |  0 2
--------------+----+----+-------+-----
o3/2o3/2x   . ♦  4 |  6 |  4  0 | 16 *
.   o3/2x4/3o ♦ 12 | 24 |  8  6 |  * 8

x3o3x *b3o

. . .    . | 32 ♦  3  3 |  3  3  3 | 3 1 1
-----------+----+-------+----------+------
x . .    . |  2 | 48  * |  2  1  0 | 2 1 0
. . x    . |  2 |  * 48 |  0  1  2 | 2 0 1
-----------+----+-------+----------+------
x3o .    . |  3 |  3  0 | 32  *  * | 1 1 0
x . x    . |  4 |  2  2 |  * 24  * | 2 0 0
. o3x    . |  3 |  0  3 |  *  * 32 | 1 0 1
-----------+----+-------+----------+------
x3o3x    . ♦ 12 | 12 12 |  4  6  4 | 8 * *
x3o . *b3o ♦  4 |  6  0 |  4  0  0 | * 8 *
. o3x *b3o ♦  4 |  0  6 |  0  0  4 | * * 8

x3o3x *b3/2o

. . .      . | 32 ♦  3  3 |  3  3  3 | 3 1 1
-------------+----+-------+----------+------
x . .      . |  2 | 48  * |  2  1  0 | 2 1 0
. . x      . |  2 |  * 48 |  0  1  2 | 2 0 1
-------------+----+-------+----------+------
x3o .      . |  3 |  3  0 | 32  *  * | 1 1 0
x . x      . |  4 |  2  2 |  * 24  * | 2 0 0
. o3x      . |  3 |  0  3 |  *  * 32 | 1 0 1
-------------+----+-------+----------+------
x3o3x      . ♦ 12 | 12 12 |  4  6  4 | 8 * *
x3o . *b3/2o ♦  4 |  6  0 |  4  0  0 | * 8 *
. o3x *b3/2o ♦  4 |  0  6 |  0  0  4 | * * 8

x3/2o3/2x *b3o

.   .   .    . | 32 ♦  3  3 |  3  3  3 | 3 1 1
---------------+----+-------+----------+------
x   .   .    . |  2 | 48  * |  2  1  0 | 2 1 0
.   .   x    . |  2 |  * 48 |  0  1  2 | 2 0 1
---------------+----+-------+----------+------
x3/2o   .    . |  3 |  3  0 | 32  *  * | 1 1 0
x   .   x    . |  4 |  2  2 |  * 24  * | 2 0 0
.   o3/2x    . |  3 |  0  3 |  *  * 32 | 1 0 1
---------------+----+-------+----------+------
x3/2o3/2x    . ♦ 12 | 12 12 |  4  6  4 | 8 * *
x3/2o   . *b3o ♦  4 |  6  0 |  4  0  0 | * 8 *
.   o3/2x *b3o ♦  4 |  0  6 |  0  0  4 | * * 8

x3/2o3/2x *b3/2o

.   .   .      . | 32 ♦  3  3 |  3  3  3 | 3 1 1
-----------------+----+-------+----------+------
x   .   .      . |  2 | 48  * |  2  1  0 | 2 1 0
.   .   x      . |  2 |  * 48 |  0  1  2 | 2 0 1
-----------------+----+-------+----------+------
x3/2o   .      . |  3 |  3  0 | 32  *  * | 1 1 0
x   .   x      . |  4 |  2  2 |  * 24  * | 2 0 0
.   o3/2x      . |  3 |  0  3 |  *  * 32 | 1 0 1
-----------------+----+-------+----------+------
x3/2o3/2x      . ♦ 12 | 12 12 |  4  6  4 | 8 * *
x3/2o   . *b3/2o ♦  4 |  6  0 |  4  0  0 | * 8 *
.   o3/2x *b3/2o ♦  4 |  0  6 |  0  0  4 | * * 8

s4x3o3o

demi( . . . . ) | 32 ♦  3  3 |  3  3  3 | 3 1 1
----------------+----+-------+----------+------
demi( . x . . ) |  2 | 48  * |  1  2  0 | 2 1 0
sefa( s4x . . ) |  2 |  * 48 |  1  0  2 | 2 0 1
----------------+----+-------+----------+------
      s4x . .   ♦  4 |  2  2 | 24  *  * | 2 0 0
demi( . x3o . ) |  3 |  3  0 |  * 32  * | 1 1 0
sefa( s4x3o . ) |  3 |  0  3 |  *  * 32 | 1 0 1
----------------+----+-------+----------+------
      s4x3o .   ♦ 12 | 12 12 |  6  4  4 | 8 * *
demi( . x3o3o ) ♦  4 |  6  0 |  0  4  0 | * 8 *
sefa( s4x3o3o ) ♦  4 |  0  6 |  0  0  4 | * * 8

s4o3o3x

demi( . . . . ) | 32 ♦  3  3 |  3  3  3 | 1 1 3
----------------+----+-------+----------+------
      s4o . .   ♦  2 | 48  * |  2  1  0 | 1 0 2
demi( . . . x ) |  2 |  * 48 |  0  1  2 | 0 1 2
----------------+----+-------+----------+------
sefa( s4o3o . ) |  3 |  3  0 | 32  *  * | 1 0 1
sefa( s4o . x ) |  4 |  2  2 |  * 24  * | 0 0 2
demi( . . o3x ) |  3 |  0  3 |  *  * 32 | 0 1 1
----------------+----+-------+----------+------
      s4o3o .   ♦  4 |  6  0 |  4  0  0 | 8 * *
demi( . o3o3x ) ♦  4 |  0  6 |  0  0  4 | * 8 *
sefa( s4o3o3x ) ♦ 12 | 12 12 |  4  6  4 | * * 8

xxoo3oxxo3ooxx&#xt   → all heights = 1/sqrt(2) = 0.707107
(tet || pseudo tut || pseudo inv tut || inv tet)

o...3o...3o...     | 4  *  * * ♦ 3  3 0  0  0  0 0  0 0 | 3 3  3 0  0  0  0 0  0 0 0 | 1 3 1 0 0 0 0
.o..3.o..3.o..     | * 12  * * ♦ 0  1 1  2  2  0 0  0 0 | 0 1  2 1  2  2  1 0  0 0 0 | 0 2 1 1 1 0 0
..o.3..o.3..o.     | *  * 12 * ♦ 0  0 0  0  2  2 1  1 0 | 0 0  0 0  1  2  2 1  2 1 0 | 0 1 0 1 2 1 0
...o3...o3...o     | *  *  * 4 ♦ 0  0 0  0  0  0 0  3 3 | 0 0  0 0  0  0  0 0  3 3 3 | 0 0 0 0 3 1 1
-------------------+-----------+------------------------+----------------------------+--------------
x... .... ....     | 2  0  0 0 | 6  * *  *  *  * *  * * | 2 1  0 0  0  0  0 0  0 0 0 | 1 2 0 0 0 0 0
oo..3oo..3oo..&#x  | 1  1  0 0 | * 12 *  *  *  * *  * * | 0 1  2 0  0  0  0 0  0 0 0 | 0 2 1 0 0 0 0
.x.. .... ....     | 0  2  0 0 | *  * 6  *  *  * *  * * | 0 1  0 0  2  0  0 0  0 0 0 | 0 2 0 1 0 0 0
.... .x.. ....     | 0  2  0 0 | *  * * 12  *  * *  * * | 0 0  1 1  0  1  0 0  0 0 0 | 0 1 1 0 1 0 0
.oo.3.oo.3.oo.&#x  | 0  1  1 0 | *  * *  * 24  * *  * * | 0 0  0 0  1  1  1 0  0 0 0 | 0 1 0 1 1 0 0
.... ..x. ....     | 0  0  2 0 | *  * *  *  * 12 *  * * | 0 0  0 0  0  1  0 1  1 0 0 | 0 1 0 0 1 1 0
.... .... ..x.     | 0  0  2 0 | *  * *  *  *  * 6  * * | 0 0  0 0  0  0  2 0  0 1 0 | 0 0 0 1 2 0 0
..oo3..oo3..oo&#x  | 0  0  1 1 | *  * *  *  *  * * 12 * | 0 0  0 0  0  0  0 0  2 1 0 | 0 0 0 0 2 1 0
.... .... ...x     | 0  0  0 2 | *  * *  *  *  * *  * 6 | 0 0  0 0  0  0  0 0  0 1 2 | 0 0 0 0 2 0 1
-------------------+-----------+------------------------+----------------------------+--------------
x...3o... ....     | 3  0  0 0 | 3  0 0  0  0  0 0  0 0 | 4 *  * *  *  *  * *  * * * | 1 1 0 0 0 0 0
xx.. .... ....&#x  | 2  2  0 0 | 1  2 1  0  0  0 0  0 0 | * 6  * *  *  *  * *  * * * | 0 2 0 0 0 0 0
.... ox.. ....&#x  | 1  2  0 0 | 0  2 0  1  0  0 0  0 0 | * * 12 *  *  *  * *  * * * | 0 1 1 0 0 0 0
.... .x..3.o..     | 0  3  0 0 | 0  0 0  3  0  0 0  0 0 | * *  * 4  *  *  * *  * * * | 0 0 1 0 1 0 0
.xo. .... ....&#x  | 0  2  1 0 | 0  0 1  0  2  0 0  0 0 | * *  * * 12  *  * *  * * * | 0 1 0 1 0 0 0
.... .xx. ....&#x  | 0  2  2 0 | 0  0 0  1  2  1 0  0 0 | * *  * *  * 12  * *  * * * | 0 1 0 0 1 0 0
.... .... .ox.&#x  | 0  1  2 0 | 0  0 0  0  2  0 1  0 0 | * *  * *  *  * 12 *  * * * | 0 0 0 1 1 0 0
..o.3..x. ....     | 0  0  3 0 | 0  0 0  0  0  3 0  0 0 | * *  * *  *  *  * 4  * * * | 0 1 0 0 0 1 0
.... ..xo ....&#x  | 0  0  2 1 | 0  0 0  0  0  1 0  2 0 | * *  * *  *  *  * * 12 * * | 0 0 0 0 1 1 0
.... .... ..xx&#x  | 0  0  2 2 | 0  0 0  0  0  0 1  2 1 | * *  * *  *  *  * *  * 6 * | 0 0 0 0 2 0 0
.... ...o3...x     | 0  0  0 3 | 0  0 0  0  0  0 0  0 3 | * *  * *  *  *  * *  * * 4 | 0 0 0 0 1 0 1
-------------------+-----------+------------------------+----------------------------+--------------
x...3o...3o...     ♦ 4  0  0 0 | 6  0 0  0  0  0 0  0 0 | 4 0  0 0  0  0  0 0  0 0 0 | 1 * * * * * *
xxo.3oxx. ....&#xt ♦ 3  6  3 0 | 3  6 3  3  6  3 0  0 0 | 1 3  3 0  3  3  0 1  0 0 0 | * 4 * * * * *
.... ox..3oo..&#x  ♦ 1  3  0 0 | 0  3 0  3  0  0 0  0 0 | 0 0  3 1  0  0  0 0  0 0 0 | * * 4 * * * *
.xo. .... .ox.&#x  ♦ 0  2  2 0 | 0  0 1  0  4  0 1  0 0 | 0 0  0 0  2  0  2 0  0 0 0 | * * * 6 * * *
.... .xxo3.oxx&#xt ♦ 0  3  6 3 | 0  0 0  3  6  3 3  6 3 | 0 0  0 1  0  3  3 0  3 3 1 | * * * * 4 * *
..oo3..xo ....&#x  ♦ 0  0  3 1 | 0  0 0  0  0  3 0  3 0 | 0 0  0 0  0  0  0 1  3 0 0 | * * * * * 4 *
...o3...o3...x     ♦ 0  0  0 4 | 0  0 0  0  0  0 0  0 6 | 0 0  0 0  0  0  0 0  0 0 4 | * * * * * * 1

or
o...3o...3o...     & | 8  * ♦  3  3  0  0  0 | 3  3  3 0  0  0 | 1 3 1 0
.o..3.o..3.o..     & | * 24 ♦  0  1  1  2  2 | 0  1  2 1  3  2 | 0 3 1 1
---------------------+------+----------------+-----------------+--------
x... .... ....     & | 2  0 | 12  *  *  *  * | 2  1  0 0  0  0 | 1 2 0 0
oo..3oo..3oo..&#x  & | 1  1 |  * 24  *  *  * | 0  1  2 0  0  0 | 0 2 1 0
.x.. .... ....     & | 0  2 |  *  * 12  *  * | 0  1  0 0  2  0 | 0 2 0 1
.... .x.. ....     & | 0  2 |  *  *  * 24  * | 0  0  1 1  0  1 | 0 2 1 0
.oo.3.oo.3.oo.&#x    | 0  2 |  *  *  *  * 24 | 0  0  0 0  2  1 | 0 2 0 1
---------------------+------+----------------+-----------------+--------
x...3o... ....     & | 3  0 |  3  0  0  0  0 | 8  *  * *  *  * | 1 1 0 0
xx.. .... ....&#x  & | 2  2 |  1  2  1  0  0 | * 12  * *  *  * | 0 2 0 0
.... ox.. ....&#x  & | 1  2 |  0  2  0  1  0 | *  * 24 *  *  * | 0 1 1 0
.... .x..3.o..     & | 0  3 |  0  0  0  3  0 | *  *  * 8  *  * | 0 1 1 0
.xo. .... ....&#x  & | 0  3 |  0  0  1  0  2 | *  *  * * 24  * | 0 1 0 1
.... .xx. ....&#x    | 0  4 |  0  0  0  2  2 | *  *  * *  * 12 | 0 2 0 0
---------------------+------+----------------+-----------------+--------
x...3o...3o...     & ♦ 4  0 |  6  0  0  0  0 | 4  0  0 0  0  0 | 2 * * *
xxo.3oxx. ....&#xt & ♦ 3  9 |  3  6  3  6  6 | 1  3  3 1  3  3 | * 8 * *
.... ox..3oo..&#x  & ♦ 1  3 |  0  3  0  3  0 | 0  0  3 1  0  0 | * * 8 *
.xo. .... .ox.&#x    ♦ 0  4 |  0  0  2  0  4 | 0  0  0 0  4  0 | * * * 6

ooo3xox4oqo&#xt   → both heights = 1/sqrt(2) = 0.707107
(co || pseudo q-cube || co)

o..3o..4o..     | 12 *  * ♦  4  2  0  0 | 2 2  4  1  0 0 0 | 1 2 2 0 0
.o.3.o.4.o.     |  * 8  * ♦  0  3  3  0 | 0 0  3  3  3 0 0 | 0 1 3 1 0
..o3..o4..o     |  * * 12 ♦  0  0  2  4 | 0 0  0  1  4 2 2 | 0 0 2 2 1
----------------+---------+-------------+------------------+----------
... x.. ...     |  2 0  0 | 24  *  *  * | 1 1  1  0  0 0 0 | 1 1 1 0 0
oo.3oo.4oo.&#x  |  1 1  0 |  * 24  *  * | 0 0  2  1  0 0 0 | 0 1 2 0 0
.oo3.oo4.oo&#x  |  0 1  1 |  *  * 24  * | 0 0  0  1  2 0 0 | 0 0 2 1 0
... ..x ...     |  0 0  2 |  *  *  * 24 | 0 0  0  0  1 1 1 | 0 0 1 1 1
----------------+---------+-------------+------------------+----------
o..3x.. ...     |  3 0  0 |  3  0  0  0 | 8 *  *  *  * * * | 1 1 0 0 0
... x..4o..     |  4 0  0 |  4  0  0  0 | * 6  *  *  * * * | 1 0 1 0 0
... xo. ...&#x  |  2 1  0 |  1  2  0  0 | * * 24  *  * * * | 0 1 1 0 0
... ... oqo&#xt |  1 2  1 |  0  2  2  0 | * *  * 12  * * * | 0 0 2 0 0
... .ox ...&#x  |  0 1  2 |  0  0  2  1 | * *  *  * 24 * * | 0 0 1 1 0
..o3..x ...     |  0 0  3 |  0  0  0  3 | * *  *  *  * 8 * | 0 0 0 1 1
... ..x4..o     |  0 0  4 |  0  0  0  4 | * *  *  *  * * 6 | 0 0 1 0 1
----------------+---------+-------------+------------------+----------
o..3x..4o..     ♦ 12 0  0 | 24  0  0  0 | 8 6  0  0  0 0 0 | 1 * * * *
oo.3xo. ...&#x  ♦  3 1  0 |  3  3  0  0 | 1 0  3  0  0 0 0 | * 8 * * *
... xox4oqo&#xt ♦  4 4  4 |  4  8  8  4 | 0 1  4  4  4 0 1 | * * 6 * *
.oo3.ox ...&#x  ♦  0 1  3 |  0  0  3  3 | 0 0  0  0  3 1 0 | * * * 8 *
..o3..x4..o     ♦  0 0 12 |  0  0  0 24 | 0 0  0  0  0 8 6 | * * * * 1

or
o..3o..4o..      & | 24 * ♦  4  2 |  2  2  4  1 | 1  2 2
.o.3.o.4.o.        |  * 8 ♦  0  6 |  0  0  6  3 | 0  2 3
-------------------+------+-------+-------------+-------
... x.. ...      & |  2 0 | 48  * |  1  1  1  0 | 1  1 1
oo.3oo.4oo.&#x   & |  1 1 |  * 48 |  0  0  2  1 | 0  1 2
-------------------+------+-------+-------------+-------
o..3x.. ...      & |  3 0 |  3  0 | 16  *  *  * | 1  1 0
... x..4o..      & |  4 0 |  4  0 |  * 12  *  * | 1  0 1
... xo. ...&#x   & |  2 1 |  1  2 |  *  * 48  * | 0  1 1
... ... oqo&#xt    |  2 2 |  0  4 |  *  *  * 12 | 0  0 2
-------------------+------+-------+-------------+-------
o..3x..4o..      & ♦ 12 0 | 24  0 |  8  6  0  0 | 2  * *
oo.3xo. ...&#x   & ♦  3 1 |  3  3 |  1  0  3  0 | * 16 *
... xox4oqo&#xt    ♦  8 4 |  8 16 |  0  2  8  4 | *  * 6

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