Acronym	spic
Name	small prismatotetracontoctachoron, runcinated icositetrachoron
Cross sections	©
Circumradius	sqrt[2+sqrt(2)] = 1.847759
Vertex figure	©
Vertex layers	Layer Symmetry Subsymmetries   o3o4o3o o3o4o . o3o . o o . o3o . o4o3o 1 x3o4o3x x3o4o . oct first x3o . x trip first x . o3x trip first . o4o3x oct first 2 x3o4x . x3q . o o . q3x . x4o3x 3 w3o4o . o3x . w w . x3o . o4o3w 4 o3x4x . w3o . x x . o3w . x4x3o 5 q3x4o . o3w . q q . w3o . o4x3q 6 o3x4x . q3x . w w . x3q . x4x3o 7 w3o4o . x3w . o o . w3x . o4o3w 8 x3o4x . o3x . X X . x3o . x4o3x 9 x3o4o . opposite oct w3x . x x . x3w . o4o3x opposite oct 10   o3w . w w . w3o   11 w3o . w w . o3w 12 x3w . x x . w3x 13 x3o . X X . o3x 14 w3x . o o . x3w 15 x3q . w w . q3x 16 w3o . q q . o3w 17 o3w . x x . w3o 18 x3o . w w . o3x 19 q3x . o o . x3q 20 o3x . x opposite trip x . x3o opposite trip (X=x+w=2x+q)
Lace city in approx. ASCII-art	©   o4o x4o o4o x4o x4x x4x x4o o4o w4o o4o x4x w4o w4o x4x x4o w4o q4q w4o x4o x4x w4o w4o x4x o4o w4o o4o x4o x4x x4x x4o o4o x4o o4o
	©   o3x o3x q3x x3o x3o o3w o3w w3o w3o x3q x3q w3x x3o x3o x3w x3w w3o w3o o3w o3w w3x w3x o3x o3x x3w q3x q3x o3w o3w w3o w3o o3x o3x x3q x3o x3o
Coordinates	(0, 0, sqrt(2)/2, (2+sqrt(2))/2)                   & all permutations, all changes of sign (1/2, 1/2, (1+sqrt(2))/2, (1+sqrt(2))/2)     & all permutations, all changes of sign
General of army	(is itself convex)
Colonel of regiment	(is itself locally convex – uniform polychoral members: by cells: oct quith sirco sroh trip siddic 48 48 0 0 0 inpac 48 0 24 0 96 spic 48 0 0 0 192 sirc 0 0 0 48 0   & others)
Confer	segmentochora: oct || sirco   uniform relative: srit   odip   related CRFs: owau prit
External links

As abstract polytope spic is isomorphic to quippic, thereby replacing oct by inverted oct resp. an antipodium vertex figure by a frustrum one.

Diminishing spic by oct-cupolae at hexadecachoral positions results in the srit. Diminishing only at one cycle of 4 of those positions results in the cyted spic.

Stott expanding an hexadecachoral subset of spic's octs results in the owau prit.

Incidence matrix according to Dynkin symbol

x3o4o3x

. . . . | 144 |   4   4 |   4   8   4 |  1  4  4  1
--------+-----+---------+-------------+------------
x . . . |   2 | 288   * |   2   2   0 |  1  2  1  0
. . . x |   2 |   * 288 |   0   2   2 |  0  1  2  1
--------+-----+---------+-------------+------------
x3o . . |   3 |   3   0 | 192   *   * |  1  1  0  0
x . . x |   4 |   2   2 |   * 288   * |  0  1  1  0
. . o3x |   3 |   0   3 |   *   * 192 |  0  0  1  1
--------+-----+---------+-------------+------------
x3o4o . ♦   6 |  12   0 |   8   0   0 | 24  *  *  *
x3o . x ♦   6 |   6   3 |   2   3   0 |  * 96  *  *
x . o3x ♦   6 |   3   6 |   0   3   2 |  *  * 96  *
. o4o3x ♦   6 |   0  12 |   0   0   8 |  *  *  * 24

or
. . . .    | 144 |   8 |   8   8 |  2   8
-----------+-----+-----+---------+-------
x . . .  & |   2 | 576 |   2   2 |  1   3
-----------+-----+-----+---------+-------
x3o . .  & |   3 |   3 | 384   * |  1   1
x . . x    |   4 |   4 |   * 288 |  0   2
-----------+-----+-----+---------+-------
x3o4o .  & ♦   6 |  12 |   8   0 | 48   *
x3o . x  & ♦   6 |   9 |   2   3 |  * 192

x3o4/3o3/2x

. .   .   . | 144 |   4   4 |   4   8   4 |  1  4  4  1
------------+-----+---------+-------------+------------
x .   .   . |   2 | 288   * |   2   2   0 |  1  2  1  0
. .   .   x |   2 |   * 288 |   0   2   2 |  0  1  2  1
------------+-----+---------+-------------+------------
x3o   .   . |   3 |   3   0 | 192   *   * |  1  1  0  0
x .   .   x |   4 |   2   2 |   * 288   * |  0  1  1  0
. .   o3/2x |   3 |   0   3 |   *   * 192 |  0  0  1  1
------------+-----+---------+-------------+------------
x3o4/3o   . ♦   6 |  12   0 |   8   0   0 | 24  *  *  *
x3o   .   x ♦   6 |   6   3 |   2   3   0 |  * 96  *  *
x .   o3/2x ♦   6 |   3   6 |   0   3   2 |  *  * 96  *
. o4/3o3/2x ♦   6 |   0  12 |   0   0   8 |  *  *  * 24

x3/2o4o3/2x

.   . .   . | 144 |   4   4 |   4   8   4 |  1  4  4  1
------------+-----+---------+-------------+------------
x   . .   . |   2 | 288   * |   2   2   0 |  1  2  1  0
.   . .   x |   2 |   * 288 |   0   2   2 |  0  1  2  1
------------+-----+---------+-------------+------------
x3/2o .   . |   3 |   3   0 | 192   *   * |  1  1  0  0
x   . .   x |   4 |   2   2 |   * 288   * |  0  1  1  0
.   . o3/2x |   3 |   0   3 |   *   * 192 |  0  0  1  1
------------+-----+---------+-------------+------------
x3/2o4o   . ♦   6 |  12   0 |   8   0   0 | 24  *  *  *
x3/2o .   x ♦   6 |   6   3 |   2   3   0 |  * 96  *  *
x   . o3/2x ♦   6 |   3   6 |   0   3   2 |  *  * 96  *
.   o4o3/2x ♦   6 |   0  12 |   0   0   8 |  *  *  * 24

xxwoqowxx3oooxxxooo4oxoxoxoxo&#xt   → height(1,2) = height(2,3) = height(4,5) = height(5,6) = height(7,8) = height(8,9) = 1/2
                                      height(3,4) = height(6,7) = 1/sqrt(2)-1/2 = 0.207107
(oct || pseudo sirco || pseudo w-oct || pseudo tic || pseudo (q,x)-toe || pseudo tic || pseudo w-oct || pseudo sirco || oct)

o........3o........4o........      & | 12  *  *  *  * |  4  4  0  0  0  0  0  0  0  0  0  0 |  4  8  4  0  0  0  0  0  0  0  0  0  0  0  0  0 | 1  4  4  1  0  0  0  0 0 0  0  0
.o.......3.o.......4.o.......      & |  * 48  *  *  * |  0  1  2  2  1  2  0  0  0  0  0  0 |  0  2  2  1  2  2  2  1  2  2  0  0  0  0  0  0 | 0  1  2  1  1  2  1  2 0 0  0  0
..o......3..o......4..o......      & |  *  * 12  *  * |  0  0  0  0  4  0  4  0  0  0  0  0 |  0  0  0  0  0  4  0  0  0  8  4  0  0  0  0  0 | 0  0  0  1  0  0  4  4 1 0  0  0
...o.....3...o.....4...o.....      & |  *  *  * 48  * |  0  0  0  0  0  2  0  2  1  2  1  0 |  0  0  0  0  0  0  1  2  2  2  0  1  2  2  2  2 | 0  0  0  0  1  1  2  2 0 1  1  2
....o....3....o....4....o....        |  *  *  *  * 24 |  0  0  0  0  0  0  2  0  0  4  0  2 |  0  0  0  0  0  0  0  0  0  4  4  0  4  2  0  2 | 0  0  0  0  0  0  4  2 1 0  1  2
-------------------------------------+----------------+-------------------------------------+-------------------------------------------------+---------------------------------
x........ ......... .........      & |  2  0  0  0  0 | 24  *  *  *  *  *  *  *  *  *  *  * |  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 1  2  1  0  0  0  0  0 0 0  0  0
oo.......3oo.......4oo.......&#x   & |  1  1  0  0  0 |  * 48  *  *  *  *  *  *  *  *  *  * |  0  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0 | 0  1  2  1  0  0  0  0 0 0  0  0
.x....... ......... .........      & |  0  2  0  0  0 |  *  * 48  *  *  *  *  *  *  *  *  * |  0  1  0  1  1  0  1  0  0  0  0  0  0  0  0  0 | 0  1  1  0  1  1  0  0 0 0  0  0
......... ......... .x.......      & |  0  2  0  0  0 |  *  *  * 48  *  *  *  *  *  *  *  * |  0  0  1  0  1  1  0  0  1  0  0  0  0  0  0  0 | 0  0  1  1  0  1  0  1 0 0  0  0
.oo......3.oo......4.oo......&#x   & |  0  1  1  0  0 |  *  *  *  * 48  *  *  *  *  *  *  * |  0  0  0  0  0  2  0  0  0  2  0  0  0  0  0  0 | 0  0  0  1  0  0  1  2 0 0  0  0
.o.o.....3.o.o.....4.o.o.....&#x   & |  0  1  0  1  0 |  *  *  *  *  * 96  *  *  *  *  *  * |  0  0  0  0  0  0  1  1  1  1  0  0  0  0  0  0 | 0  0  0  0  1  1  1  1 0 0  0  0
..o.o....3..o.o....4..o.o....&#x   & |  0  0  1  0  1 |  *  *  *  *  *  * 48  *  *  *  *  * |  0  0  0  0  0  0  0  0  0  2  2  0  0  0  0  0 | 0  0  0  0  0  0  2  1 1 0  0  0
......... ...x..... .........      & |  0  0  0  2  0 |  *  *  *  *  *  *  * 48  *  *  *  * |  0  0  0  0  0  0  0  1  0  0  0  1  1  0  1  0 | 0  0  0  0  1  0  1  0 0 1  0  1
......... ......... ...x.....      & |  0  0  0  2  0 |  *  *  *  *  *  *  *  * 24  *  *  * |  0  0  0  0  0  0  0  0  2  0  0  0  0  2  0  0 | 0  0  0  0  0  1  0  2 0 0  1  0
...oo....3...oo....4...oo....&#x   & |  0  0  0  1  1 |  *  *  *  *  *  *  *  *  * 96  *  * |  0  0  0  0  0  0  0  0  0  1  0  0  1  1  0  1 | 0  0  0  0  0  0  1  1 0 0  1  1
...o.o...3...o.o...4...o.o...&#x     |  0  0  0  2  0 |  *  *  *  *  *  *  *  *  *  * 24  * |  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2  2 | 0  0  0  0  0  0  0  0 0 1  1  2
......... ....x.... .........        |  0  0  0  0  2 |  *  *  *  *  *  *  *  *  *  *  * 24 |  0  0  0  0  0  0  0  0  0  0  2  0  2  0  0  0 | 0  0  0  0  0  0  2  0 1 0  0  1
-------------------------------------+----------------+-------------------------------------+-------------------------------------------------+---------------------------------
x........3o........ .........      & |  3  0  0  0  0 |  3  0  0  0  0  0  0  0  0  0  0  0 | 16  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 1  1  0  0  0  0  0  0 0 0  0  0
xx....... ......... .........&#x   & |  2  2  0  0  0 |  1  2  1  0  0  0  0  0  0  0  0  0 |  * 48  *  *  *  *  *  *  *  *  *  *  *  *  *  * | 0  1  1  0  0  0  0  0 0 0  0  0
......... ......... ox.......&#x   & |  1  2  0  0  0 |  0  2  0  1  0  0  0  0  0  0  0  0 |  *  * 48  *  *  *  *  *  *  *  *  *  *  *  *  * | 0  0  1  1  0  0  0  0 0 0  0  0
.x.......3.o....... .........      & |  0  3  0  0  0 |  0  0  3  0  0  0  0  0  0  0  0  0 |  *  *  * 16  *  *  *  *  *  *  *  *  *  *  *  * | 0  1  0  0  1  0  0  0 0 0  0  0
.x....... ......... .x.......      & |  0  4  0  0  0 |  0  0  2  2  0  0  0  0  0  0  0  0 |  *  *  *  * 24  *  *  *  *  *  *  *  *  *  *  * | 0  0  1  0  0  1  0  0 0 0  0  0
......... ......... .xo......&#x   & |  0  2  1  0  0 |  0  0  0  1  2  0  0  0  0  0  0  0 |  *  *  *  *  * 48  *  *  *  *  *  *  *  *  *  * | 0  0  0  1  0  0  0  1 0 0  0  0
.x.o..... ......... .........&#x   & |  0  2  0  1  0 |  0  0  1  0  0  2  0  0  0  0  0  0 |  *  *  *  *  *  * 48  *  *  *  *  *  *  *  *  * | 0  0  0  0  1  1  0  0 0 0  0  0
......... .o.x..... .........&#x   & |  0  1  0  2  0 |  0  0  0  0  0  2  0  1  0  0  0  0 |  *  *  *  *  *  *  * 48  *  *  *  *  *  *  *  * | 0  0  0  0  1  0  1  0 0 0  0  0
......... ......... .x.x.....&#x   & |  0  2  0  2  0 |  0  0  0  1  0  2  0  0  1  0  0  0 |  *  *  *  *  *  *  *  * 48  *  *  *  *  *  *  * | 0  0  0  0  0  1  0  1 0 0  0  0
.oooo....3.oooo....4.oooo....&#xr  & |  0  1  1  1  1 |  0  0  0  0  1  1  1  0  0  1  0  0 |  *  *  *  *  *  *  *  *  * 96  *  *  *  *  *  * | 0  0  0  0  0  0  1  1 0 0  0  0
......... ..o.x.... .........&#x   & |  0  0  1  0  2 |  0  0  0  0  0  0  2  0  0  0  0  1 |  *  *  *  *  *  *  *  *  *  * 48  *  *  *  *  * | 0  0  0  0  0  0  1  0 1 0  0  0
...o.....3...x..... .........      & |  0  0  0  3  0 |  0  0  0  0  0  0  0  3  0  0  0  0 |  *  *  *  *  *  *  *  *  *  *  * 16  *  *  *  * | 0  0  0  0  1  0  0  0 0 1  0  0
......... ...xx.... .........&#x   & |  0  0  0  2  2 |  0  0  0  0  0  0  0  1  0  2  0  1 |  *  *  *  *  *  *  *  *  *  *  *  * 48  *  *  * | 0  0  0  0  0  0  1  0 0 0  0  1
......... ......... ...xo....&#x   & |  0  0  0  2  1 |  0  0  0  0  0  0  0  0  1  2  0  0 |  *  *  *  *  *  *  *  *  *  *  *  *  * 48  *  * | 0  0  0  0  0  0  0  1 0 0  1  0
......... ...x.x... .........&#x     |  0  0  0  4  0 |  0  0  0  0  0  0  0  2  0  0  2  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  * 24  * | 0  0  0  0  0  0  0  0 0 1  0  1
...ooo...3...ooo...4...ooo...&#xr    |  0  0  0  2  1 |  0  0  0  0  0  0  0  0  0  2  1  0 |  *  *  *  *  *  *  *  *  *  *  *  *  *  *  * 48 | 0  0  0  0  0  0  0  0 0 0  1  1
-------------------------------------+----------------+-------------------------------------+-------------------------------------------------+---------------------------------
x........3o........4o........      & ♦  6  0  0  0  0 | 12  0  0  0  0  0  0  0  0  0  0  0 |  8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 | 2  *  *  *  *  *  *  * * *  *  *
xx.......3oo....... .........&#x   & ♦  3  3  0  0  0 |  3  3  3  0  0  0  0  0  0  0  0  0 |  1  3  0  1  0  0  0  0  0  0  0  0  0  0  0  0 | * 16  *  *  *  *  *  * * *  *  *
xx....... ........  ox.......&#x   & ♦  2  4  0  0  0 |  1  4  2  2  0  0  0  0  0  0  0  0 |  0  2  2  0  1  0  0  0  0  0  0  0  0  0  0  0 | *  * 24  *  *  *  *  * * *  *  *
......... ooo......4oxo......&#xt  & ♦  1  4  1  0  0 |  0  4  0  4  4  0  0  0  0  0  0  0 |  0  0  4  0  0  4  0  0  0  0  0  0  0  0  0  0 | *  *  * 12  *  *  *  * * *  *  *
.x.o.....3.o.x..... .........&#x   & ♦  0  3  0  3  0 |  0  0  3  0  0  6  0  3  0  0  0  0 |  0  0  0  1  0  0  3  3  0  0  0  1  0  0  0  0 | *  *  *  * 16  *  *  * * *  *  *
.x.o..... ......... .x.x.....&#x   & ♦  0  4  0  2  0 |  0  0  2  2  0  4  0  0  1  0  0  0 |  0  0  0  0  1  0  2  0  2  0  0  0  0  0  0  0 | *  *  *  *  * 24  *  * * *  *  *
......... .ooxx.... .........&#xr  & ♦  0  1  1  2  2 |  0  0  0  0  1  2  2  1  0  2  0  1 |  0  0  0  0  0  0  0  1  0  2  1  0  1  0  0  0 | *  *  *  *  *  * 48  * * *  *  *
......... ......... .xoxo....&#xr  & ♦  0  2  1  2  1 |  0  0  0  1  2  2  1  0  1  2  0  0 |  0  0  0  0  0  1  0  0  1  2  0  0  0  1  0  0 | *  *  *  *  *  *  * 48 * *  *  *
......... ..o.x.o..4..o.o.o..&#xt    ♦  0  0  2  0  4 |  0  0  0  0  0  0  8  0  0  0  0  4 |  0  0  0  0  0  0  0  0  0  0  8  0  0  0  0  0 | *  *  *  *  *  *  *  * 6 *  *  *
...o.o...3...x.x... .........&#x     ♦  0  0  0  6  0 |  0  0  0  0  0  0  0  6  0  0  3  0 |  0  0  0  0  0  0  0  0  0  0  0  2  0  0  3  0 | *  *  *  *  *  *  *  * * 8  *  *
...oqo... ......... ...xox...&#xt    ♦  0  0  0  4  2 |  0  0  0  0  0  0  0  0  2  8  2  0 |  0  0  0  0  0  0  0  0  0  0  0  0  0  4  0  4 | *  *  *  *  *  *  *  * * * 12  *
......... ...xxx... .........&#xr    ♦  0  0  0  4  2 |  0  0  0  0  0  0  0  2  0  4  2  1 |  0  0  0  0  0  0  0  0  0  0  0  0  2  0  1  2 | *  *  *  *  *  *  *  * * *  * 24