# User:Cube26 Cube26

## My questions

### A4+ symmetry classification

If Hemidodecahedron (4-dimensional) has space: 4D Euclidean space and A4+ symmetry is the symmetry group of Hemidodecahedron (4-dimensional), is A4+ symmetry the 4D Euclidean symmetry group?

For those who are currently using a mobile phone to edit things in these links, I have a special heading for many special links that cannot be accessed normally.

## My new notation for anisogons (My intention to extend the notion of a star polygon)

Let $r_{0} = 1, r_{1},...,r_{n} \in (1,+\infty)\,$ and $0\leq\alpha_{0},\alpha_{1},...,\alpha_{n} \leq 2\pi\,$ be "the ratio between the length of nth edge and the first one" and the "nth interior angle of the anisogon (measured in radian)".

Then $(\{r_{0},r_{1},...,r_{n}\},\{\alpha_{0},\alpha_{1},...,\alpha_{n}\})$ notates an anisogon produced by the following actions:

- Put the center $A_{1}$ and $A_{2}$ in 2D Cartesian plane.

- Draw a line from $A_{1}$ to $A_{2}$ .

+ Let $i=1$ .

+ Repeat the following actions but if at any step, there's a cyclic graph, terminate.

• Find $A_{i+2}\,$ such that$\frac{|A_{i+1}A_{i+2}|}{|A_{1}A_{2}|}=r_{i\,(\text{mod}\,n+1)}\,$ and $\angle A_{i}A_{i+1}A_{i+2}=\alpha_{i-1\,(\text{mod}\,n+1)}$ .
• Draw a line from $A_{i+1}\,$ to $A_{i+2}$ .
• Increment $i$ ### Note

Note that this method only depends on ratio between edge length so we need a convention on how to find and read the coordinates.

The convention is $A_{1}=(0,0)\,$ and $A_{2}=(1,0)\,$ as coordinates.

## My new anisogon

### Vertex coordinates

The vertices of the anisogon $(\{1,2\},\{\frac{\pi}{2},\frac{2\pi}{3}\})$ , centered at the origin has the following vertices:

• $(0,0)$ • $(1,0)$ • $(1,2)$ • $\Big(\frac{2-\sqrt{3}}{2},\frac{5}{2}\Big)$ • $\Big(-\frac{\sqrt{3}}{2},\frac{5-2\sqrt{3}}{2}\Big)$ • $\Big(\frac{1-\sqrt{3}}{2},\frac{5-3\sqrt{3}}{2}\Big)$ • $\Big(\frac{1+\sqrt{3}}{2},\frac{7-3\sqrt{3}}{2}\Big)$ • $\Big(\frac{1+\sqrt{3}}{2},\frac{9-3\sqrt{3}}{2}\Big)$ • $\Big(\frac{\sqrt{3}-3}{2},\frac{9-3\sqrt{3}}{2}\Big)$ • $\Big(\frac{\sqrt{3}-4}{2},\frac{9-4\sqrt{3}}{2}\Big)$ • $\Big(\frac{3\sqrt{3}-4}{2},\frac{7-4\sqrt{3}}{2}\Big)$ • $\Big(2\sqrt{3}-2,4-2\sqrt{3}\Big)$ • $\Big(2\sqrt{3}-3,4-\sqrt{3}\Big)$ • $\Big(2\sqrt{3}-4,4-\sqrt{3}\Big)$ • $\Big(2\sqrt{3}-4,2-\sqrt{3}\Big)$ • $\Big(\frac{5\sqrt{3}-8}{2},\frac{3-2\sqrt{3}}{2}\Big)$ • $\Big(\frac{5\sqrt{3}-6}{2},\frac{3}{2}\Big)$ • $\Big(\frac{5\sqrt{3}-7}{2},\frac{3+\sqrt{3}}{2}\Big)$ • $\Big(\frac{3\sqrt{3}-7}{2},\frac{1+\sqrt{3}}{2}\Big)$ • $\Big(\frac{3\sqrt{3}-7}{2},\frac{\sqrt{3}-1}{2}\Big)$ • $\Big(\frac{3\sqrt{3}-3}{2},\frac{\sqrt{3}-1}{2}\Big)$ • $\Big(\frac{3\sqrt{3}-2}{2},\frac{2\sqrt{3}-1}{2}\Big)$ • $\Big(\frac{\sqrt{3}-2}{2},\frac{2\sqrt{3}+1}{2}\Big)$ • $\Big(-1,\sqrt{3}\Big)$ ## This notation for some polygon

Name My notation
Star pentambus $(\{1,1\},\{\frac{7\pi}{5},\frac{\pi}{5}\})$ Pentambus $(\{1,1\},\{\alpha,\frac{8\pi}{5}-\alpha\})$ Dipentagon $(\{1,a\},\{\frac{4\pi}{5},\frac{4\pi}{5}\})$ Distellagram $(\{1,a\},\{\frac{2\pi}{5},\frac{2\pi}{5}\})\,$ where $a>\frac{1+\sqrt{5}}{2}$ Stellapod $(\{1,a\},\{\frac{2\pi}{5},\frac{2\pi}{5}\})\,$ where $1 Complex dipentagon (Degenerate) $(\{1,\frac{1+\sqrt{5}}{2}\},\{\frac{2\pi}{5},\frac{2\pi}{5}\})\,$ or $(\{1,\frac{1+\sqrt{5}}{2}\},\{\frac{\pi}{5},\frac{\pi}{5}\})\,$ Dipentagram $(\{1,a\},\{\frac{3\pi}{5},\frac{3\pi}{5}\})\,$ Distellagon $(\{1,a\},\{\frac{\pi}{5},\frac{\pi}{5}\})\,$ where $a<\frac{1+\sqrt{5}}{2}$ Pentagram $(\{1\},\{\frac{\pi}{5}\})\,$ Pentapod $(\{1,a\},\{\frac{\pi}{5},\frac{\pi}{5}\})\,$ where $a>\frac{1+\sqrt{5}}{2}$ Bowtie $(\{1,a\},\{\text{arccos}\Big(\frac{1}{a}\Big),\text{arccos}\Big(\frac{1}{a}\Big)\})\,$ Ditetragon $(\{1,a\},\{\frac{3\pi}{4},\frac{3\pi}{4}\})\,$ Regular n-gon $(\{1\},\{\frac{(n-2)\pi}{n}\})\,$ Ditetragram $(\{1,a\},\{\frac{\pi}{4},\frac{\pi}{4}\})\,$ Ditrigon $(\{1,a\},\{\frac{2\pi}{3},\frac{2\pi}{3}\})\,$ Tripod $(\{1,a\},\{\frac{\pi}{3},\frac{π}{3}\})\,$ where $1 Propeller tripod $(\{1,a\},\{\frac{\pi}{3},\frac{π}{3}\})\,$ where $a>2$ Hemitripod $(\{1,2\},\{\frac{\pi}{3},\frac{π}{3}\})\,$ Rectangle $(\{1,a\},\{\frac{\pi}{2},\frac{π}{2}\})\,$ Square $(\{1\},\{\frac{\pi}{2}\})\,$ 