example mg
(also has half-turn rotational symmetry)
(image created
in Kali and
adjusted in IrfanView)
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Groups / Symmetry patterns |
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Page background is the wallpaper pattern p3ml (image created in Kali and adjusted in IrfanView).
Border or
Frieze Patterns
There are
7 different types of border pattern. For each type, the symmetries (transformations
that result in an identical pattern) form an infinite transformation group.
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11 1m 1g 12 m1 mm mg |
| first symbol | second symbol |
| 1 - no vertical reflection | 1 - no other symmetry |
| m - vertical reflection | m - horizontal reflection |
| g - glide reflection | |
| 2 - half turn rotation |
example mg
(also has half-turn rotational symmetry)
(image created
in Kali and
adjusted in IrfanView)
A more detailed
version of this notation is available:
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This notation consists of four symbols pxyz:
| step |  |
| hop |  |
| jump |  |
| sidle |  |
| spinhop |  |
| spinjump |  |
| spinsidle |  |
John Conway
is also responsible for the orbifold notation (^^, ^*, ...) that is used
in the Kali
software and Kali
Java applet. The following table shows all four notations. In the Kali
Java applet, the ^ symbol is replaced with the infinity sign and x is replaced
by o.
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In the orbifold notation:
frieze
applet - Frank Farris
Border
Pattern Gallery - Oklahoma State University
class project creating and classifying
border patterns
ten
friezes (border patterns)
Which ones are mathematically the
same?
two
colour friezes
Slavik Jablan's Symmetry and Ornament
includes a diagram showing the 17 possible frieze patterns when two colours
are permitted - 1 step, 1 hop, 3 jump, 2 sidle, 2 spinhop, 5 spinjump and
3 spinsidle.
frieze/border
images
The
Art of Oriental Carpets - Plane or Field Patterns - Math Forum
17
wallpaper groups - David Hop
Animations effectively demonstrate
the transformations that result in identical patterns.
17
wallpaper tilings - Geometry Centre
descriptions and animations
17
wallpaper groups - Wikipedia
descriptions
17
wallpaper patterns - Escher
Do these tessellations by Escher
represent all 17 possible types?
Japanese
patterns
Traditional Japanese patterns are
used to provide examples of the different types of wallpaper pattern (except
for p3). The Japanese names for the patterns are given as well as a detailed
analysis.
examples
of the 17 wallpaper patterns - Hans Kuiper
Donald Duck included!
examples
of the 17 wallpaper patterns - Helmer Aslaksen
Images
of Chaos and Symmetry (Quilts) - Professor Mike Field
Vibrating
Wallpaper - Frank Farris
This paper provides movies of wallpaper
patterns that continuously transform themselves.
poster
of the 17 wallpaper patterns - Andrew Crompton
Each pattern is made with flying
creatures.
Tiling
Plane and Fancy - Steve Edwards
detailed analysis of the 17 different
plane symmetry groups with a large number of examples taken from many different
cultures
Wallpaper
Groups (Plane Symmetry Groups) - David Joyce
detailed analysis of the 17 different
plane symmetry groups
Plane
Crystallographic Groups - Phill Schultz
detailed analysis of the 17 different
plane symmetry groups
Wallpaper
Patterns - Xah Lee
description
of the 17 wallpaper patterns - David Reid
Symmetries
of Culture - Donald Crowe
Dror
Bar-Natan's page
Symmetry
and the Shape of Space - Chaim Goodman-Strauss
Mat
103 (Symmetries) - George Baloglou
course on symmetry of plane with
examples, descriptions, problems, ...
Symmetry
and Crystals - Steven Dutch
An analysis of crystals is preceded
by an exploration of symmetry around a point in the plane, repeating patterns
on a strip (one-dimensional space groups), repeating patterns in the plane
(two-dimensional space groups), symmetry around a point in three dimensions
(3-dimensional point groups) and repeating patterns in space (three-dimensional
space groups).
symmetry
explored in terms of group notation
wallpaper
images
Protozone
- Al Jarnow
Escher
Web Sketch - Java applet
Kali
- Java applet
Create patterns
based on symmetry groups - 11 rosette, 7 frieze and 17 wallpaper groups
- can be downloaded.
Kali
software
Create patterns
based on symmetry groups - 11 rosette, 7 frieze and 17 wallpaper groups.
Tess
software
Create patterns
based on symmetry groups - 11 rosette, 7 frieze and 17 wallpaper groups.
Introduction
to Group Theory - Dog School of Mathematics
introduction, Cayley tables, symmetry
group of equilateral triangle, subgroups, cyclic groups, permutation groups,
Rubik's cube
Groups
- Peter Williams
notes including an excellent interactive
demonstration of the symmetry
group of the square
symmetry
groups Java applet- Barbara De La Rue & Frances Goldman
Investigate the symmetry transformations
of regular polygons and stars.
Abstract
Algebra Online - John Beachy
abstract algebra course for undergraduate
students including multiplication
tables for groups of small order
Group
Theory - J.S.Milne
undergraduate notes - pdf file
Groups
of small order - John Pederson
Diamond
Puzzle - Steven Cullinane
This puzzle is an example of a permutation
group in geometry with 322560 elements. The 2x2
case has 24 elements.
Development
of group theory - MacTutor History Archive
The
abstract group concept - MacTutor History Archive
Rubik's
Cube Group - W. D. Joyner
lecture notes on the mathematics
of the Rubik's cube
Groups15
The Java applet that lets you work
with groups of order 1 to 15
Groups32
CayleyQuilter
- software
Create a mathematical mosaic based
on a Cayley table, then reflect and rotate it to make quilt-like patterns.
The patterns can be saved as bitmap files and they make interesting background
tiles on desktops and web pages. (Teaching
resources by Laurie Burton.)
Finite
Group Behaviour - software
Work with existing groups or create
your own. Test for associativity and inverses. Create subgroups.
Group
Explorer - software
View finite groups as Cayley
diagrams and symmetry objects.
PascalGT
- software
Group theory meets Pascal's triangle
- a PascGalois triangle. Construct Pascal's triangle but, instead of standard
integer addition, use a group binary operation to determine the entries.
Symmetry
and Group Theory - collection of links - Geometry Junkyard
resources
page - symmetry
additional material including links
for rosette patterns - plaster rosettes and medallions, wheel trims, snowflakes,
mandalas, Kirigami
