Tesseract and 16-cell faces

This list shows the k-faces of the tesseract and its dual 16-cell.


16-cell projection
8 vertices
24 edges (forming 6 rings)
32 triangular faces (forming 4 spheres)
graph corresponding to 15 of 16 cells


ternary weight 0 1 2 3 4
number of faces 0 8 24 32 16
tesseract faces t 4
tesseract
t 3
cube
t 2
square
t 1
edge
t 0
vertex
16-cell faces c 4
16-cell
c 0
vertex
c 1
edge
c 2
triangle
c 3
tetrahedron

The sign vectors (with entries 1, 0, −1) in the following table are the face centers of the tesseract. (Compare example for the cube.)
The indices on the left (between −40 and 40) are their interpretation as little-endian balanced ternary numbers.
The columns to their right are their sums, patterns of non-zero entries, and Hamming weights (numbers of non-zero entries).
The default order of the table is first by Hamming weights, then by patterns, and then by sums.


index b. t.
vector
s p w tesseract
face type
tesseract
vertices
tesseract
image
16-cell
face type
16-cell
image
0 0 0
....
0 t 4
tesseract
!!!! !!!! !!!! !!!! c 4
16-cell
−1 −1 1
!...
1 t 3
cube
!.!. !.!. !.!. !.!. c 0
vertex
1 1 1
!...
1 t 3
cube
.!.! .!.! .!.! .!.! c 0
vertex
−3 −1 2
.!..
1 t 3
cube
!!.. !!.. !!.. !!.. c 0
vertex
3 1 2
.!..
1 t 3
cube
..!! ..!! ..!! ..!! c 0
vertex
−9 −1 4
..!.
1 t 3
cube
!!!! .... !!!! .... c 0
vertex
9 1 4
..!.
1 t 3
cube
.... !!!! .... !!!! c 0
vertex
−27 −1 8
...!
1 t 3
cube
!!!! !!!! .... .... c 0
vertex
27 1 8
...!
1 t 3
cube
.... .... !!!! !!!! c 0
vertex
−4 −2 3
!!..
2 t 2
square
!... !... !... !... c 1
edge
−2 0 3
!!..
2 t 2
square
.!.. .!.. .!.. .!.. c 1
edge
2 0 3
!!..
2 t 2
square
..!. ..!. ..!. ..!. c 1
edge
4 2 3
!!..
2 t 2
square
...! ...! ...! ...! c 1
edge
−10 −2 5
!.!.
2 t 2
square
!.!. .... !.!. .... c 1
edge
−8 0 5
!.!.
2 t 2
square
.!.! .... .!.! .... c 1
edge
8 0 5
!.!.
2 t 2
square
.... !.!. .... !.!. c 1
edge
10 2 5
!.!.
2 t 2
square
.... .!.! .... .!.! c 1
edge
−12 −2 6
.!!.
2 t 2
square
!!.. .... !!.. .... c 1
edge
−6 0 6
.!!.
2 t 2
square
..!! .... ..!! .... c 1
edge
6 0 6
.!!.
2 t 2
square
.... !!.. .... !!.. c 1
edge
12 2 6
.!!.
2 t 2
square
.... ..!! .... ..!! c 1
edge
−28 −2 9
!..!
2 t 2
square
!.!. !.!. .... .... c 1
edge
−26 0 9
!..!
2 t 2
square
.!.! .!.! .... .... c 1
edge
26 0 9
!..!
2 t 2
square
.... .... !.!. !.!. c 1
edge
28 2 9
!..!
2 t 2
square
.... .... .!.! .!.! c 1
edge
−30 −2 10
.!.!
2 t 2
square
!!.. !!.. .... .... c 1
edge
−24 0 10
.!.!
2 t 2
square
..!! ..!! .... .... c 1
edge
24 0 10
.!.!
2 t 2
square
.... .... !!.. !!.. c 1
edge
30 2 10
.!.!
2 t 2
square
.... .... ..!! ..!! c 1
edge
−36 −2 12
..!!
2 t 2
square
!!!! .... .... .... c 1
edge
−18 0 12
..!!
2 t 2
square
.... !!!! .... .... c 1
edge
18 0 12
..!!
2 t 2
square
.... .... !!!! .... c 1
edge
36 2 12
..!!
2 t 2
square
.... .... .... !!!! c 1
edge
−13 −3 7
!!!.
3 t 1
edge
0, 8 c 2
triangle
−11 −1 7
!!!.
3 t 1
edge
1, 9 c 2
triangle
−7 −1 7
!!!.
3 t 1
edge
2, 10 c 2
triangle
5 −1 7
!!!.
3 t 1
edge
4, 12 c 2
triangle
−5 1 7
!!!.
3 t 1
edge
3, 11 c 2
triangle
7 1 7
!!!.
3 t 1
edge
5, 13 c 2
triangle
11 1 7
!!!.
3 t 1
edge
6, 14 c 2
triangle
13 3 7
!!!.
3 t 1
edge
7, 15 c 2
triangle
−31 −3 11
!!.!
3 t 1
edge
0, 4 c 2
triangle
−29 −1 11
!!.!
3 t 1
edge
1, 5 c 2
triangle
−25 −1 11
!!.!
3 t 1
edge
2, 6 c 2
triangle
23 −1 11
!!.!
3 t 1
edge
8, 12 c 2
triangle
−23 1 11
!!.!
3 t 1
edge
3, 7 c 2
triangle
25 1 11
!!.!
3 t 1
edge
9, 13 c 2
triangle
29 1 11
!!.!
3 t 1
edge
10, 14 c 2
triangle
31 3 11
!!.!
3 t 1
edge
11, 15 c 2
triangle
−37 −3 13
!.!!
3 t 1
edge
0, 2 c 2
triangle
−35 −1 13
!.!!
3 t 1
edge
1, 3 c 2
triangle
−19 −1 13
!.!!
3 t 1
edge
4, 6 c 2
triangle
17 −1 13
!.!!
3 t 1
edge
8, 10 c 2
triangle
−17 1 13
!.!!
3 t 1
edge
5, 7 c 2
triangle
19 1 13
!.!!
3 t 1
edge
9, 11 c 2
triangle
35 1 13
!.!!
3 t 1
edge
12, 14 c 2
triangle
37 3 13
!.!!
3 t 1
edge
13, 15 c 2
triangle
−39 −3 14
.!!!
3 t 1
edge
0, 1 c 2
triangle
−33 −1 14
.!!!
3 t 1
edge
2, 3 c 2
triangle
−21 −1 14
.!!!
3 t 1
edge
4, 5 c 2
triangle
15 −1 14
.!!!
3 t 1
edge
8, 9 c 2
triangle
−15 1 14
.!!!
3 t 1
edge
6, 7 c 2
triangle
21 1 14
.!!!
3 t 1
edge
10, 11 c 2
triangle
33 1 14
.!!!
3 t 1
edge
12, 13 c 2
triangle
39 3 14
.!!!
3 t 1
edge
14, 15 c 2
triangle
−40 −4 15
!!!!
4 t 0
vertex
0 c 3
tetrahedron
−38 −2 15
!!!!
4 t 0
vertex
1 c 3
tetrahedron
−34 −2 15
!!!!
4 t 0
vertex
2 c 3
tetrahedron
−22 −2 15
!!!!
4 t 0
vertex
4 c 3
tetrahedron
14 −2 15
!!!!
4 t 0
vertex
8 c 3
tetrahedron
−32 0 15
!!!!
4 t 0
vertex
3 c 3
tetrahedron
−20 0 15
!!!!
4 t 0
vertex
5 c 3
tetrahedron
−16 0 15
!!!!
4 t 0
vertex
6 c 3
tetrahedron
16 0 15
!!!!
4 t 0
vertex
9 c 3
tetrahedron
20 0 15
!!!!
4 t 0
vertex
10 c 3
tetrahedron
32 0 15
!!!!
4 t 0
vertex
12 c 3
tetrahedron
−14 2 15
!!!!
4 t 0
vertex
7 c 3
tetrahedron
22 2 15
!!!!
4 t 0
vertex
11 c 3
tetrahedron
34 2 15
!!!!
4 t 0
vertex
13 c 3
tetrahedron
38 2 15
!!!!
4 t 0
vertex
14 c 3
tetrahedron
40 4 15
!!!!
4 t 0
vertex
15 c 3
tetrahedron