File:Fundamental Domains and Symmetry Groups for Uniform Tilings and Polyhedrons.png
Jump to navigation
Jump to search
Size of this preview: 800 × 267 pixels. Other resolutions: 320 × 107 pixels | 640 × 213 pixels | 1,024 × 341 pixels | 1,280 × 427 pixels | 4,800 × 1,600 pixels.
Original file (4,800 × 1,600 pixels, file size: 209 KB, MIME type: image/png)
Captions
Captions
Add a one-line explanation of what this file represents
Summary
[edit]
| Description |
English: Illustration of the Fundamental Domains of point groups in polyhedrons and planar tilings. The four groups are D_3, D_4, D_5, and D_6, the dihedral groups of degree 3,4,5,6, respectively. These groups are formed as the semidirect products of incremental rotations and reflections. |
| Date | |
| Source | Own work |
| Author | Harry Princeton |
Licensing
[edit]
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
[asy]
real l=1/6;
pair c=(0,0);
for(int i=0; i<3; ++i)
{
filldraw(c--c+expi(pi/2+i*2*pi/3)/(2*sin(pi/3))--c+expi(pi/2+pi/3+i*2*pi/3)/(2*tan(pi/3))--cycle,rgb(1,1/2,1/2),black+linewidth(l));
draw(c+expi(pi/2+pi/3+i*2*pi/3)/(2*tan(pi/3))--c+expi(pi/2+2*pi/3+i*2*pi/3)/(2*sin(pi/3)),black+linewidth(l));
}
label("\textcolor{red}{Triangular}",c-(0,1)*(1/(2*tan(pi/3))+1/4));
c=(4/3,0);
for(int i=0; i<4; ++i)
{
filldraw(c--c+expi(pi/4+i*2*pi/4)/(2*sin(pi/4))--c+expi(pi/4+pi/4+i*2*pi/4)/(2*tan(pi/4))--cycle,rgb(1/2,1/2,1),black+linewidth(l));
draw(c+expi(pi/4+pi/4+i*2*pi/4)/(2*tan(pi/4))--c+expi(pi/4+2*pi/4+i*2*pi/4)/(2*sin(pi/4)),black+linewidth(l));
}
label("\textcolor{blue}{Square}",c-(0,1)*(1/(2*tan(pi/4))+1/4));
c=(3,0);
for(int i=0; i<5; ++i)
{
filldraw(c--c+expi(pi/2+i*2*pi/5)/(2*sin(pi/5))--c+expi(pi/2+pi/5+i*2*pi/5)/(2*tan(pi/5))--cycle,rgb(1/2,1,1/2),black+linewidth(l));
draw(c+expi(pi/2+pi/5+i*2*pi/5)/(2*tan(pi/5))--c+expi(pi/2+2*pi/5+i*2*pi/5)/(2*sin(pi/5)),black+linewidth(l));
}
label("\textcolor{green}{Pentagonal}",c-(0,1)*(1/(2*tan(pi/5))+1/4));
c=(26/5,0);
for(int i=0; i<6; ++i)
{
filldraw(c--c+expi(i*2*pi/6)/(2*sin(pi/6))--c+expi(pi/6+i*2*pi/6)/(2*tan(pi/6))--cycle,rgb(1/2,1/2,1/2),black+linewidth(l));
draw(c+expi(pi/6+i*2*pi/6)/(2*tan(pi/6))--c+expi(2*pi/6+i*2*pi/6)/(2*sin(pi/6)),black+linewidth(l));
}
label("Hexagonal",c-(0,1)*(1/(2*tan(pi/6))+1/4));
[/asy]
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 20:35, 19 October 2019 | | 4,800 × 1,600 (209 KB) | Harry Princeton (talk | contribs) | Changed file to a Asymptote (LaTeX) version. |
| 16:15, 19 October 2019 |  | 3,200 × 943 (67 KB) | Harry Princeton (talk | contribs) | Labeled chambers of fundamental domains correctly. If polyhedra are chiral, then colored chambers are mapped only to colored chambers. | |
| 17:03, 10 July 2019 |  | 3,200 × 943 (77 KB) | Harry Princeton (talk | contribs) | Cross-wiki upload from commons.wikimedia.org |
You cannot overwrite this file.
File usage on Commons
The following 2 pages use this file:
