# 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?

## Some special links

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 ${\displaystyle r_{0} = 1, r_{1},...,r_{n} \in (1,+\infty)\,}$and ${\displaystyle 0\leq\alpha_{0},\alpha_{1},...,\alpha_{n} \leq 2\pi\,}$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 ${\displaystyle (\{r_{0},r_{1},...,r_{n}\},\{\alpha_{0},\alpha_{1},...,\alpha_{n}\})}$ notates an anisogon produced by the following actions:

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

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

+ Let ${\displaystyle i=1}$.

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

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

### 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 ${\displaystyle A_{1}=(0,0)\,}$and ${\displaystyle A_{2}=(1,0)\,}$as coordinates.

## My new anisogon

### Vertex coordinates

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

• ${\displaystyle (0,0)}$
• ${\displaystyle (1,0)}$
• ${\displaystyle (1,2)}$
• ${\displaystyle \Big(\frac{2-\sqrt{3}}{2},\frac{5}{2}\Big)}$
• ${\displaystyle \Big(-\frac{\sqrt{3}}{2},\frac{5-2\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(\frac{1-\sqrt{3}}{2},\frac{5-3\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(\frac{1+\sqrt{3}}{2},\frac{7-3\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(\frac{1+\sqrt{3}}{2},\frac{9-3\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(\frac{\sqrt{3}-3}{2},\frac{9-3\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(\frac{\sqrt{3}-4}{2},\frac{9-4\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(\frac{3\sqrt{3}-4}{2},\frac{7-4\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(2\sqrt{3}-2,4-2\sqrt{3}\Big)}$
• ${\displaystyle \Big(2\sqrt{3}-3,4-\sqrt{3}\Big)}$
• ${\displaystyle \Big(2\sqrt{3}-4,4-\sqrt{3}\Big)}$
• ${\displaystyle \Big(2\sqrt{3}-4,2-\sqrt{3}\Big)}$
• ${\displaystyle \Big(\frac{5\sqrt{3}-8}{2},\frac{3-2\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(\frac{5\sqrt{3}-6}{2},\frac{3}{2}\Big)}$
• ${\displaystyle \Big(\frac{5\sqrt{3}-7}{2},\frac{3+\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(\frac{3\sqrt{3}-7}{2},\frac{1+\sqrt{3}}{2}\Big)}$
• ${\displaystyle \Big(\frac{3\sqrt{3}-7}{2},\frac{\sqrt{3}-1}{2}\Big)}$
• ${\displaystyle \Big(\frac{3\sqrt{3}-3}{2},\frac{\sqrt{3}-1}{2}\Big)}$
• ${\displaystyle \Big(\frac{3\sqrt{3}-2}{2},\frac{2\sqrt{3}-1}{2}\Big)}$
• ${\displaystyle \Big(\frac{\sqrt{3}-2}{2},\frac{2\sqrt{3}+1}{2}\Big)}$
• ${\displaystyle \Big(-1,\sqrt{3}\Big)}$

## This notation for some polygon

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