# Tesseract and 16-cell faces

This list shows the ${\displaystyle 3^{4}=81}$ 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

Searching first by weight and then by pattern gives a meaningful order.
Both terms refer to the non-zero entries of the little-endian balanced ternary vector on the left.
These vectors are the coordinate vectors of the face centers of the tesseract. (Compare example for the cube.)
A map from faces (−40 to 40) to tesseract vertices (0 to 15) can be found here.

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