Symmetries of Four Dimensions - Part 5

This page will list the symmetries of objects in four dimensions. It will deal with polytopes (as well as round objects) but not include the tilings and honeycomb symmetries. These symmetries are often called point groups due to being centered on one point, the origin. I will introduce some 'short names' for the symmetries along with a Coxeter notation for it. I'll extend Coxeter's notation to include round symmetries also. This page will link to other pages which detail various symmetries and the isogonal (vertex-transitive) and isotopal (facet-transitive aka dice) shapes there. Part 5 will take on the swirlcubic symmetries.

Swirlcubic Symmetries

1-Scubic Symmetry = 2-Skicubic Symmetry - [4,3:1] order 96.

2-Scubic Symmetry - [4,3:2] order 192.

3-Scubic Symmetry - [4,3:3] order 288.

4-Scubic Symmetry - [4,3:4] order 384.

5-Scubic Symmetry - [4,3:5] order 480.

6-Scubic Symmetry - [4,3:6] order 576.

7-Scubic Symmetry - [4,3:7] order 672.

8-Scubic Symmetry - [4,3:8] order 768.

9-Scubic Symmetry - [4,3:9] order 864.

10-Scubic Symmetry - [4,3:10] order 960.

12-Scubic Symmetry - [4,3:12] order 1152.

15-Scubic Symmetry - [4,3:15] order 1440.

16-Scubic Symmetry - [4,3:16] order 1536.

18-Scubic Symmetry - [4,3:18] order 1728.

20-Scubic Symmetry - [4,3:20] order 1920.

24-Scubic Symmetry - [4,3:24] order 2304.

Swirlkicubic Symmetries

1-Skicubic Symmetry - [4,3:1]⁺ order 48.

2-Skicubic Symmetry = 1-Scubic Symmetry - [4,3:2]⁺ order 96.

3-Skicubic Symmetry - [4,3:3]⁺ order 144.

4-Skicubic Symmetry - [4,3:4]⁺ order 192.

5-Skicubic Symmetry - [4,3:5]⁺ order 240.

6-Skicubic Symmetry - [4,3:6]⁺ order 288.

7-Skicubic Symmetry - [4,3:7]⁺ order 336.

8-Skicubic Symmetry - [4,3:8]⁺ order 384.

9-Skicubic Symmetry - [4,3:9]⁺ order 432.

10-Skicubic Symmetry - [4,3:10]⁺ order 480.

12-Skicubic Symmetry - [4,3:12]⁺ order 576.

15-Skicubic Symmetry - [4,3:15]⁺ order 720.

16-Skicubic Symmetry - [4,3:16]⁺ order 768.

18-Skicubic Symmetry - [4,3:18]⁺ order 864.

20-Skicubic Symmetry - [4,3:20]⁺ order 960.

24-Skicubic Symmetry - [4,3:24]⁺ order 1152.

Altswirlcubic Symmetries

1-Ascubic Symmetry - [4;3:1] order 96.

2-Ascubic Symmetry - [4;3:2] order 192.

3-Ascubic Symmetry - [4;3:3] order 288.

4-Ascubic Symmetry - [4;3:4] order 384.

5-Ascubic Symmetry - [4;3:5] order 480.

6-Ascubic Symmetry - [4;3:6] order 576.

7-Ascubic Symmetry - [4;3:7] order 672.

8-Ascubic Symmetry - [4;3:8] order 768.

9-Ascubic Symmetry - [4;3:9] order 864.

10-Ascubic Symmetry - [4;3:10] order 960.

12-Ascubic Symmetry - [4;3:12] order 1152.

15-Ascubic Symmetry - [4;3:15] order 1440.

16-Ascubic Symmetry - [4;3:16] order 1536.

18-Ascubic Symmetry - [4;3:18] order 1728.

20-Ascubic Symmetry - [4;3:20] order 1920.

24-Ascubic Symmetry - [4;3:24] order 2304.

Altswirlkicubic Symmetries

1-Askicubic Symmetry - [4;3:1]⁺ order 48.

2-Askicubic Symmetry - [4;3:2]⁺ order 96.

3-Askicubic Symmetry - [4;3:3]⁺ order 144.

4-Askicubic Symmetry - [4;3:4]⁺ order 192.

5-Askicubic Symmetry - [4;3:5]⁺ order 240.

6-Askicubic Symmetry - [4;3:6]⁺ order 288.

7-Askicubic Symmetry - [4;3:7]⁺ order 336.

8-Askicubic Symmetry - [4;3:8]⁺ order 384.

9-Askicubic Symmetry - [4;3:9]⁺ order 432.

10-Askicubic Symmetry - [4;3:10]⁺ order 480.

12-Askicubic Symmetry - [4;3:12]⁺ order 576.

15-Askicubic Symmetry - [4;3:15]⁺ order 720.

16-Askicubic Symmetry - [4;3:16]⁺ order 768.

18-Askicubic Symmetry - [4;3:18]⁺ order 864.

20-Askicubic Symmetry - [4;3:20]⁺ order 960.

24-Askicubic Symmetry - [4;3:24]⁺ order 1152.

Swirlpyritic Symmetries

1-Spyritic Symmetry - [4,3⁺:1] order 48.

2-Spyritic Symmetry - [4,3⁺:2] order 96.

3-Spyritic Symmetry - [4,3⁺:3] order 144.

4-Spyritic Symmetry - [4,3⁺:4] order 192.

5-Spyritic Symmetry - [4,3⁺:5] order 240.

6-Spyritic Symmetry - [4,3⁺:6] order 288.

7-Spyritic Symmetry - [4,3⁺:7] order 336.

8-Spyritic Symmetry - [4,3⁺:8] order 384.

9-Spyritic Symmetry - [4,3⁺:9] order 432.

10-Spyritic Symmetry - [4,3⁺:10] order 480.

12-Spyritic Symmetry - [4,3⁺:12] order 576.

15-Spyritic Symmetry - [4,3⁺:15] order 720.

16-Spyritic Symmetry - [4,3⁺:16] order 768.

18-Spyritic Symmetry - [4,3⁺:18] order 864.

20-Spyritic Symmetry - [4,3⁺:20] order 960.

24-Spyritic Symmetry - [4,3⁺:24] order 1152.

Four Dimensional Symmetries - Part 4 - Swirltettic Symmetries . . . Four Dimensional Symmetries - Part 6 - Duoprismic Symmetries

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