ex

Acronym ex

Name hexacosachoron,
600-cell,
hydrochoron

Cross sections

©

Circumradius (1+sqrt(5))/2 = 1.618034

Inradius sqrt[(9+4 sqrt(5))/8] = 1.497676

Vertex figure

©

Vertex layers

Layer	Symmetry	Subsymmetries
	`o3o3o5o`	`o3o3o .`	`o3o . o`	`o . o5o`	`. o3o5o`
1	`x3x3o5o`	`x3o3o .` tet first	`x3o . o` {3} first	`x . o5o` edge first	`. o3o5o` vertex first
2		`o3o3f .`	`o3o . f`	`o . o5x`	`. x3o5o` vertex figure
3		`o3f3o .`	`o3f . x`	`f . x5o`	`. o3o5x`
4		`f3o3x .`	`f3o . f`	`F . o5o`	`. f3o5o`
5		`o3x3f .`	`x3f . o`	`o . f5o`	`. o3x5o`
6		`x3f3o .`	`o3x . F`	`x . o5f`	`. f3o5o`
7		`F3o3o .`	`o3F . o`	`F . o5x`	`. o3o5x`
8		`f3o3f .`	`f3x . f`	`f . f5o`	`. x3o5o`
9a		`o3o3F .`	`F3o . x`	`o . x5x`	`. o3o5o` opposite vertex
9b		`o3o3F .`	`F3o . x`	`V . o5o`	`. o3o5o` opposite vertex
10		`o3f3x .`	`f3o . F`	`f . o5f`
11a		`f3x3o .`	`f3f . o`	`F . x5o`
11b		`f3x3o .`	`o3o . V`	`F . x5o`
12		`x3o3f .`	`o3f . F`	`x . f5o`
13		`o3f3o .`	`o3F . x`	`o . o5f`
14		`f3o3o .`	`x3f . f`	`F . o5o`
15		`o3o3x .` opposite tet	`F3o . o`	`f . o5x`
16			`x3o . F`	`o . x5o`
17			`f3x . o`	`x . o5o` opposite edge
18			`o3f . f`
19			`f3o . x`
20			`o3o . f`
21			`o3x . o` opposite {3}

(F=ff=f+x=2x+v, V=F+v=2x+2v=2f)

Lace city
in approx. ASCII-art

©	o5o o5o o5x x5o x5o o5o o5o f5o o5f o5f o5x o5x f5o f5o o5o x5x o5o o5f o5f x5o x5o f5o f5o o5f o5o o5o o5x o5x x5o o5o o5o

©	o3o x3o o3f f3o o3x o3o o3f f3x x3f f3o o3o f3o o3F F3o o3f o3x x3f F3o f3f o3F f3x x3o f3o o3F F3o o3f o3o o3f f3x x3f f3o o3o x3o o3f f3o o3x o3o

Coordinates

(τ, 0, 0, 0) & all permutations, all changes of sign
(vertex inscribed f/q-hex)
(τ/2, τ/2, τ/2, τ/2) & all permutations, all changes of sign
(vertex inscribed f-tes)
(τ²/2, τ/2, 1/2, 0) & even permutations, all changes of sign
(vertex inscribed sadi)

where τ = (1+sqrt(5))/2 (a. and b. together define a vertex inscribed f-ico)

General of army (is itself convex)

Colonel of regiment

(is itself locally convex – uniform polychoral members:

by cells:	doe	gad	ike	tet
gahi	120	0	0	0
gohi	0	120	0	0
fix	0	0	120	0
ex	0	0	0	600

& others)

Dual hi

Confer

Grünbaumian relatives:: 2ex 2ex+120ike ex+fix ex+gohi+120id
segmentochora:: ikepy ike || doe doe || id
other CRFs:: 1/10-luna of ex 2/10-luna of ex 3/10-luna of ex 4/10-luna of ex bidex id || f-ike || ike
uniform relative:: sadi
general polytopal classes:: tetrahedrochora regular

External
links

As abstract polytope ex is isomorphic to gax, thereby replacing pentagonal edge figures by pentagrammal ones resp. replacing ike vertex figures by gike ones.

Diminishing ex at the vertex positions of an inscribed f-ico results in sadi. – Diminishing likewise at the vertex positions of 2 inscribed f-icoes, which mutually are not vertex incident, results in bidex. – Double-diminishing ex at the vertex positions of an inscribed f/q-hex, i.e. chopping off (intersecting) bistratic caps, results in f-tes.

Incidence matrix according to Dynkin symbol

x3o3o5o

. . . . | 120 ♦  12 |   30 |  20
--------+-----+-----+------+----
x . . . |   2 | 720 |    5 |   5
--------+-----+-----+------+----
x3o . . |   3 |   3 | 1200 |   2
--------+-----+-----+------+----
x3o3o . ♦   4 |   6 |    4 | 600

x3o3o5/4o

. . .   . | 120 ♦  12 |   30 |  20
----------+-----+-----+------+----
x . .   . |   2 | 720 |    5 |   5
----------+-----+-----+------+----
x3o .   . |   3 |   3 | 1200 |   2
----------+-----+-----+------+----
x3o3o   . ♦   4 |   6 |    4 | 600

x3o3/2o5o

. .   . . | 120 ♦  12 |   30 |  20
----------+-----+-----+------+----
x .   . . |   2 | 720 |    5 |   5
----------+-----+-----+------+----
x3o   . . |   3 |   3 | 1200 |   2
----------+-----+-----+------+----
x3o3/2o . ♦   4 |   6 |    4 | 600

x3o3/2o5/4o

. .   .   . | 120 ♦  12 |   30 |  20
------------+-----+-----+------+----
x .   .   . |   2 | 720 |    5 |   5
------------+-----+-----+------+----
x3o   .   . |   3 |   3 | 1200 |   2
------------+-----+-----+------+----
x3o3/2o   . ♦   4 |   6 |    4 | 600

x3/2o3o5o

.   . . . | 120 ♦  12 |   30 |  20
----------+-----+-----+------+----
x   . . . |   2 | 720 |    5 |   5
----------+-----+-----+------+----
x3/2o . . |   3 |   3 | 1200 |   2
----------+-----+-----+------+----
x3/2o3o . ♦   4 |   6 |    4 | 600

x3/2o3o5/4o

.   . .   . | 120 ♦  12 |   30 |  20
------------+-----+-----+------+----
x   . .   . |   2 | 720 |    5 |   5
------------+-----+-----+------+----
x3/2o .   . |   3 |   3 | 1200 |   2
------------+-----+-----+------+----
x3/2o3o   . ♦   4 |   6 |    4 | 600

x3/2o3/2o5o

.   .   . . | 120 ♦  12 |   30 |  20
------------+-----+-----+------+----
x   .   . . |   2 | 720 |    5 |   5
------------+-----+-----+------+----
x3/2o   . . |   3 |   3 | 1200 |   2
------------+-----+-----+------+----
x3/2o3/2o . ♦   4 |   6 |    4 | 600

x3/2o3/2o5/4o

.   .   .   . | 120 ♦  12 |   30 |  20
--------------+-----+-----+------+----
x   .   .   . |   2 | 720 |    5 |   5
--------------+-----+-----+------+----
x3/2o   .   . |   3 |   3 | 1200 |   2
--------------+-----+-----+------+----
x3/2o3/2o   . ♦   4 |   6 |    4 | 600

oxofofoxo3ooooxoooo5ooxoooxoo&#xt   → height(1,2) = height(3,4) = height(6,7) = height(8,9) = (sqrt(5)-1)/4 = 0.309017
                                      height(2,3) = height(4,5) = height(5,6) = height(7,8) = 1/2
(pt || pseudo ike || pseudo doe || pseudo f-ike || pseudo id || pseudo f-ike || pseudo doe || pseudo ike || pt)

o........3o........5o........      & | 2  *  *  *  * ♦ 12  0   0  0  0   0   0  0   0  0 | 30  0   0   0   0   0   0  0   0   0  0  0 | 20  0  0   0  0   0   0  0
.o.......3.o.......5.o.......      & | * 24  *  *  * ♦  1  5   5  1  0   0   0  0   0  0 |  5  5  10   5   5   0   0  0   0   0  0  0 |  5  5  5   5  0   0   0  0
..o......3..o......5..o......      & | *  * 40  *  * ♦  0  0   3  0  3   3   3  0   0  0 |  0  0   3   6   3   6   3  3   6   0  0  0 |  0  1  3   6  1   3   6  0
...o.....3...o.....5...o.....      & | *  *  * 24  * ♦  0  0   0  1  0   5   0  1   5  0 |  0  0   0   0   5   5   0  0  10   5  5  0 |  0  0  0   5  0   5   5  5
....o....3....o....5....o....        | *  *  *  * 30 ♦  0  0   0  0  0   0   4  0   4  4 |  0  0   0   0   0   0   8  2   8   8  2  2 |  0  0  0   0  4   8   4  4
-------------------------------------+---------------+-----------------------------------+--------------------------------------------+---------------------------
oo.......3oo.......5oo.......&#x   & | 1  1  0  0  0 | 24  *   *  *  *   *   *  *   *  * |  5  0   0   0   0   0   0  0   0   0  0  0 |  5  0  0   0  0   0   0  0
.x....... ......... .........      & | 0  2  0  0  0 |  * 60   *  *  *   *   *  *   *  * |  1  2   2   0   0   0   0  0   0   0  0  0 |  2  2  1   0  0   0   0  0
.oo......3.oo......5.oo......&#x   & | 0  1  1  0  0 |  *  * 120  *  *   *   *  *   *  * |  0  0   2   2   1   0   0  0   0   0  0  0 |  0  1  2   2  0   0   0  0
.o.o.....3.o.o.....5.o.o.....&#x   & | 0  1  0  1  0 |  *  *   * 24  *   *   *  *   *  * |  0  0   0   0   5   0   0  0   0   0  0  0 |  0  0  0   5  0   0   0  0
......... ......... ..x......      & | 0  0  2  0  0 |  *  *   *  * 60   *   *  *   *  * |  0  0   0   2   0   2   0  1   0   0  0  0 |  0  0  1   2  0   0   2  0
..oo.....3..oo.....5..oo.....&#x   & | 0  0  1  1  0 |  *  *   *  *  * 120   *  *   *  * |  0  0   0   0   1   2   0  0   2   0  0  0 |  0  0  0   2  0   1   2  0
..o.o....3..o.o....5..o.o....&#x   & | 0  0  1  0  1 |  *  *   *  *  *   * 120  *   *  * |  0  0   0   0   0   0   2  1   2   0  0  0 |  0  0  0   0  1   2   2  0
...o.o...3...o.o...5...o.o...&#x     | 0  0  0  2  0 |  *  *   *  *  *   *   * 12   *  * |  0  0   0   0   0   0   0  0   0   0  5  0 |  0  0  0   0  0   0   0  5
...oo....3...oo....5...oo....&#x   & | 0  0  0  1  1 |  *  *   *  *  *   *   *  * 120  * |  0  0   0   0   0   0   0  0   2   2  1  0 |  0  0  0   0  0   2   1  2
......... ....x.... .........        | 0  0  0  0  2 |  *  *   *  *  *   *   *  *   * 60 |  0  0   0   0   0   0   2  0   0   2  0  1 |  0  0  0   0  2   2   0  1
-------------------------------------+---------------+-----------------------------------+--------------------------------------------+---------------------------
ox....... ......... .........&#x   & | 1  2  0  0  0 |  2  1   0  0  0   0   0  0   0  0 | 60  *   *   *   *   *   *  *   *   *  *  * |  2  0  0   0  0   0   0  0
.x.......3.o....... .........      & | 0  3  0  0  0 |  0  3   0  0  0   0   0  0   0  0 |  * 40   *   *   *   *   *  *   *   *  *  * |  1  1  0   0  0   0   0  0
.xo...... ......... .........&#x   & | 0  2  1  0  0 |  0  1   2  0  0   0   0  0   0  0 |  *  * 120   *   *   *   *  *   *   *  *  * |  0  1  1   0  0   0   0  0
......... ......... .ox......&#x   & | 0  1  2  0  0 |  0  0   2  0  1   0   0  0   0  0 |  *  *   * 120   *   *   *  *   *   *  *  * |  0  0  1   1  0   0   0  0
.ooo.....3.ooo.....5.ooo.....&#x   & | 0  1  1  1  0 |  0  0   1  1  0   1   0  0   0  0 |  *  *   *   * 120   *   *  *   *   *  *  * |  0  0  0   2  0   0   0  0
......... ......... ..xo.....&#x   & | 0  0  2  1  0 |  0  0   0  0  1   2   0  0   0  0 |  *  *   *   *   * 120   *  *   *   *  *  * |  0  0  0   1  0   0   1  0
......... ..o.x.... .........&#x   & | 0  0  1  0  2 |  0  0   0  0  0   0   2  0   0  1 |  *  *   *   *   *   * 120  *   *   *  *  * |  0  0  0   0  1   1   0  0
......... ......... ..x.o....&#x   & | 0  0  2  0  1 |  0  0   0  0  1   0   2  0   0  0 |  *  *   *   *   *   *   * 60   *   *  *  * |  0  0  0   0  0   0   2  0
..ooo....3..ooo....5..ooo....&#x   & | 0  0  1  1  1 |  0  0   0  0  0   1   1  0   1  0 |  *  *   *   *   *   *   *  * 240   *  *  * |  0  0  0   0  0   1   1  0
......... ...ox.... .........&#x   & | 0  0  0  1  2 |  0  0   0  0  0   0   0  0   2  1 |  *  *   *   *   *   *   *  *   * 120  *  * |  0  0  0   0  0   1   0  1
...ooo...3...ooo...5...ooo...&#x     | 0  0  0  2  1 |  0  0   0  0  0   0   0  1   2  0 |  *  *   *   *   *   *   *  *   *   * 60  * |  0  0  0   0  0   0   0  2
....o....3....x.... .........        | 0  0  0  0  3 |  0  0   0  0  0   0   0  0   0  3 |  *  *   *   *   *   *   *  *   *   *  * 20 |  0  0  0   0  2   0   0  0
-------------------------------------+---------------+-----------------------------------+--------------------------------------------+---------------------------
ox.......3oo....... .........&#x   & ♦ 1  3  0  0  0 |  3  3   0  0  0   0   0  0   0  0 |  3  1   0   0   0   0   0  0   0   0  0  0 | 40  *  *   *  *   *   *  *
.xo......3.oo...... .........&#x   & ♦ 0  3  1  0  0 |  0  3   3  0  0   0   0  0   0  0 |  0  1   3   0   0   0   0  0   0   0  0  0 |  * 40  *   *  *   *   *  *
.xo...... ......... .ox......&#x   & ♦ 0  2  2  0  0 |  0  1   4  0  1   0   0  0   0  0 |  0  0   2   2   0   0   0  0   0   0  0  0 |  *  * 60   *  *   *   *  *
......... ......... .oxo.....&#xt  & ♦ 0  1  2  1  0 |  0  0   2  1  1   2   0  0   0  0 |  0  0   0   1   2   1   0  0   0   0  0  0 |  *  *  * 120  *   *   *  *
..o.o....3..o.x.... .........&#x   & ♦ 0  0  1  0  3 |  0  0   0  0  0   0   3  0   0  3 |  0  0   0   0   0   0   3  0   0   0  0  1 |  *  *  *   * 40   *   *  *
......... ..oox.... .........&#xt  & ♦ 0  0  1  1  2 |  0  0   0  0  0   1   2  0   2  1 |  0  0   0   0   0   0   1  0   2   1  0  0 |  *  *  *   *  * 120   *  *
......... ......... ..xoo....&#xt  & ♦ 0  0  2  1  1 |  0  0   0  0  1   2   2  0   1  0 |  0  0   0   0   0   1   0  1   2   0  0  0 |  *  *  *   *  *   * 120  *
......... ...oxo... .........&#xt    ♦ 0  0  0  2  2 |  0  0   0  0  0   0   0  1   4  1 |  0  0   0   0   0   0   0  0   0   2  2  0 |  *  *  *   *  *   *   * 60

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