Number 9 Cut the knot

A Journey into CyberSpace
Geometry on the Internet.
Compiled by Belisha J Price (Mathematics Teacher Cambridge H.S.) during his 2006 Sabbatical
I have set this document up in 5 sections…
Section A Polyhedra
Section B Flag Construction
Section C Origamic Architecture
Section D Paper models
Section E Bibliography
Section A Polyhedra
Having surfed thousands of websites pertaining to Geometry I have chosen these as sites that I feel
could be of interest to anyone who enjoys Geometry and would like to explore the subject further. I
have a particular interest in Polyhedra and that is where my emphasis lies. Some of the sites in this
section are in BOLD TYPE and these I feel are particularly worth a visit. I have also put others in
LARGER BOLD TYPE and these are the sites that I found more interesting and I have spent much
more time on them. There are some sites in this Section that deal with other aspects of Mathematics
and I thought that they would be worth including.
Section B Flag Construction
These Flag Construction sites are from the Flags of the World website and show how precise the
construction of National Flags is. The most complex one is the flag of Nepal and this is an excellent
test of a student’s ability to follow instructions and use the necessary geometry equipment.
By the way the whole of the Flags of the World website is full of the History and the Geography of
the countries of the world. It is well worth spending some time on this site.
Section C Origamic Architecture
The Origamic Architecture( Pop-ups) are a good way of exploring 3 dimensions and, in Japan,
they are used by Architectural students at University level. Books by Prof Chatani give excellent
value for money.
Section D Paper models
The Paper Models are both cheap, (many are free downloads) and are great fun for the students to
do. The variety of models on the web that are available is amazing.
Section E Bibliography
This is just a small selection of books that are available and that I use in my own studies on
I gather that this year (2006) is the Web’s 15th year in operation and it is truly amazing to see what is
on there and the rate at which it is growing. I have only scratched the surface of available Geometry
sites and will continue to look for more sites that are of use to anyone that has an interest in the
Section A
My Number 1 site is emphasized as, in my opinion, it is the best site and has literally millions of Polyhedra
and their nets all downloadable. The software is very user friendly and Robert Webb regularly upgrades the
software. There is a limited free trial download to try. The software is certainly a lot cheaper than the books
that I have on the subject of Polyhedra. Anyhow this short list of 9 websites would make a good starting
point for viewing and, hopefully, will whet your appetite to look through the numbered list that follows.
By being on line and clicking on one of the websites in this document you should be able to access
the site without having to type in the address. If this fails to work then just copy and paste the site in the
address bar of your server window. But in general one click will take you to the site.
I hope you can find something that is of interest and that you can use in your teaching programme.
Number 1 Great Stella
This is THE NUMBER ONE SITE for Polyhedra Software.
I have, over the years, spent a small fortune on books on Polyhedra and in comparison this software
is Cheap. If you are serious about Polyhedra … Buy.
Number 2 Non Orthogonal Polyhedra
An interesting but rather technical PDF.
Number 3 Build Your Own Polyhedra
This site has 19 Polyhedra. They can be printed as GIFs or PSs. By using PS and opening in
Coreldraw you are able to colour individual polygons and to change the size of the resulting models.
There is also reference to a Windows programme called Flatland.
Number 4 Waterman models
The Waterman polyhedra are the result of packing spheres and this site illustrates this very well.
Great graphics.
Number 5 Polyhedra
This is a book review.
Number 6 Polyhedra Pastimes
This site is a must. There are many interesting things to do and make. There are a lot of useful
commercial materials to buy and there are some book reviews. The Investigating Patterns series of
books published by Dale Seymour looks a good series and the links on the site are coordinated with
the activities and chapters in the second book in the series.
Number 7 Tessellated Polyhedra
5 neat Polyhedra with Escher type patterns on them. These nets can be printed off.
Number 8 The 4 Regular non-convex polyheda
An interesting dissertation on the Keplar-Poinsot models with reference to the Golden Section.
Number 9 Cut the knot
A very good website covering all areas of maths including puzzles and puzzle theory, and
book reviews. There are a number of definitions, theorems and proofs.
Number 10 Regular Polyhedra made with Magz.
This site illustrates the relationship between the various basic polyhedra and would make a good
project for a class to construct the complete set.
Number 11 Polyhedra Posters
Two colourful posters on Polyhedra 22”x34” (Commercial)
Number 12 Sierpinski Poyhedra
A 10 page PDF on Sierpinski Polyhedra with illustrations and theory.
Numbers 13 and 14 Jorge Rezende
This is a Portuguese website which has Polyhedra puzzles in PDF format. The site can be viewed in
Number 15 Modular Origami
This site shows the technique for making polyhedra from modular pieces.
Number 16 Numericana
A very interesting site with a great deal of information. I literally spent hours on this site after
clicking on Index. There is so much material of interest. Many related links.
Number 17 Polyhedra
A site with some interesting Polyhedra.
Number 18 Mathematics Polyhedra
A simple site that gives some description and PDF’s for some basic polyhedra.
Number 19 Polyhedra Links
Some very good polyhedra links.
Number 20 Origami Polyhedra
A very good site on modular origami. Students could work in teams to create some of the models.
Number 21 Studying Polyhedra
An applet on the Platonics.
Number 22 Modular Origami Web resources
Number 23 Wolfram Polyhedra
This is an excellent website from a very much bigger site. The Wolfram website is absolutely
fascinating and has great width and depth. It is a MUST site.
Number 24 Wolfram Uniform Polyhedra
This part of the Wolfram site gives a great deal of information the on 80 Uniform Polyhedra and their
Number 25 Martin Trump’s Polyhedra
This is an interesting programme that uses an applet to illustrate Martin’s polyhedra and method.
Unfortunately it does not give angle sizes but these can be calculated from the lengths given.
Number 26 An Introduction to Golden Rhombic Polyhedra
The faces of all these polyhedra have the 2 diagonals that are in Golden Section. There are only 5
convex examples but there are many others in the group.
Number 27 Associated with Number 26
An interesting development of the Rhombic Polyhedra.
Number 28 Origami Poyhedra
This site leads to a number of interesting sites on Origami Polyhedra.
Number 29 Impossible Polyhedra
Contains an interesting proof and some good links to other ideas such as mazes, knots and Magic
Number 30 Epcot Ball (Origami)
Basic instructions to build Origami polyhedra.
Number 31 Polyhedra Mathmos site
This site has 55 models (5 platonics, 4 Kepler-Poinsot, 13 Archimedeans, 13 Archimedean Duals and
20 Stellated Archimedeans) giving details and calculations.
Number 32 a set of PDFs
This is a site with some fairly heavy maths covering a wide range of ideas. It’s not for the faint
Number 33 Greek Mathematics
Some mathematical history. Worth a visit.
Number 34 Rhombic Polyhedra
This website is a must. There is a huge amount of polyhedra on which are superimposed maps of the
world and they make a fascinating set to make. There are also maps of Cook’s Journeys on models.
Perhaps another class project.
Number 35 - 36 Archimedean
Interesting sites that illustrate the truncations of basic polyhedra. Good illustrations. Worth a visit.
Number 37 -38 Graziotti and Fleurent
These sites have plates that would make very good classroom posters. There is also other very
interesting material.
Number 39 - 41 Waterman Packing spheres
These sites show the results of Waterman’s research over a number of years.
Number 42 - 43 Polyhedra
These sites have a wide variety of aspects to do with polyhedra. The site uses VRML to show
Number 44 3D Puzzles
This site is under construction but looks as though it will have material worth looking at.
Number 45 Beyond Flatland
Pascal’s Triangle etc. This is Flatland the software not the book that is referred to in Number 73.
Number 46 A Ride Through the Polyhedra World
A site that uses many applets to illustrate and inform about Polyhedra. Some interesting extra notes.
Number 47 Answers
This site gathers information from the many dictionaries and encyclopedias on line and gives very
detailed information about all sorts of polyhedra and their families. A very good reference site.
Number 48 Posters
This site has a series of Polyhedra Posters for sale and there are booklets with the posters.
Number 49 Polyhedra Applets
A well illustrated site with the Platonics, the Kepler-Poinsot solids and a good explanation of
Number 50 BBC Mathematics
An interactive site covering a number of Maths areas. Could be useful for junior or low ability
Number 51 Bonner Design
This site shows models that are decorated with Islamic patterns. It could inspire the more artistic
Number 52 Polyhedra Software
I have not used this software but it is reasonably priced and may be useful.
Number 53 Centre for Innovation in Mathematics Teaching
I think that this site would be well worth exploring.
Number 54 A Guide to Sources of Mathematics on the Internet
An excellent source of information with many other websites to visit.
Number 55 CoolMath4Kids
A site designed for kids and is bright, attractive and user friendly. Kids would enjoy using this site.
Number 56 Unfolding Models to Nets
An animated site that shows the unfolding of many shapes into nets. This includes all the
pentominoes. Perhaps another project for group work.
Number 57 PDF
This PDF is rather technical but I include it for those that are interested in theory.
Number 58 and 59 Jim Planks Origami page
Modular models.
Number 60 Gergonne’s Magic
A lot of interesting tricks and puzzles. Worth looking at.
Number 61 Cut the Knot
This site is full of information on Polyhedra and also has a lot of puzzles.
Number 62 Euclid’s Elements on the Platonics
Geometry as it used to be. A nice bit of history and mathematical discipline.
Number 63 Enchanted Learning
This site covers all areas of the Curriculum not just Mathematics. In the Mathematics side it covers
everything and this page gives basic polyhedra and historical notes.
Number 64 Platonic and Archimedean Polyhedra
This site covers the basic 18 models and also leads to other theorems etc
Number 65 Plaited Polyhedra
A different approach to making Polyhedra. This technique is easy and downloadable patterns are
Number 66 KidsKount
An interactive website filled with all sorts of puzzles and games. Kids would love this one.
Number 67 Woven Polyhedra
A rather advanced technique but one that is worth looking at. The enlarged images are quite amazing
to see. Mainly geodesics.
Number 68 Platonic and Archimedean Polyhedra
Another site with basic information presented in a slightly different way. It has links to George
Hart’s website which is a must to visit.
Number 69 Platonics and the Soccerball
There are interesting other parts to visit on this site.
Number 70 Geodesic Maths Links
There are some very good links here.
Number 71 Polyhedra
Much theory and calculation but there are other interesting pages to follow through.
Number 72 George Hart
This site is a must. George Hart is a Sculptor as well as a mathematician. His site has a whole host
of valuable information etc. He also shows his own works which are quite inspirational.
Number 73 Flatland the book
This is a PDF of the book Flatland – A Romance of many dimensions. The book was first published
in 1884. An 82 page book, which makes interesting reading. To Quote Wikipedia
The story
The story posits a two dimensional world (Flatland). The narrator, a humble square (named A. Square), guides us through some of
the implications of life in two dimensions. A. Square dreams of a visit to a one-dimensional world (Lineland), and attempts to
convince the realm's ignorant monarch of a second dimension. The narrator is then visited by a three-dimensional sphere, which he
cannot comprehend until he sees the third dimension for himself. He then dreams of visiting Pointland (which comprises a selfaware point that occupies all space and knows nothing but itself) with the Sphere and learns that he cannot "rescue [the point]
from his self-satisfaction". He learns to aspire and to teach others to aspire. The role of women is explained, along with a class
system, both of which are a satire of Victorian society at the time .
Social elements
In the book, men are portrayed as polygons whose social class is directly proportional to the number of sides they have; therefore,
triangles, having only three sides, are at the bottom of the social ladder and are considered generally unintelligent, while the
Priests are composed of multisided polygons whose shapes approximate a circle, which is considered to be the "perfect" shape. On
the other hand, the female population is comprised only of lines, who are required by law to sway back and forth and sound a
"peace-cry" as they walk, due to the fact that when a line is coming towards an observer in a 2-D world, it appears merely as a
point. A. Square talks of accounts where men have been killed (both by accident and on purpose) by being stabbed by women.
This explains the need for separate doors for women and men in buildings.
In the world of Flatland, classes are distinguished using the "Art of Feeling" and the "Art of Sight Recognition". Feeling, practiced
by the lower classes and women, determines the configuration of a person by feeling one of their angles. The "Art of Sight
Recognition", practiced by the upper classes, is aided by "Fog", which allows an observer to determine the depth of an object.
With this, polygons with sharp angles relative to the observer will fade out more rapidly than polygons with more gradual angles.
The population of Flatland can "evolve" through the Law of Nature, which states:
"a male child shall have one more side than his father, so that each generation shall rise (as a rule) one step in the scale
of development and nobility. Thus the son of a Square is a Pentagon; the son of a Pentagon, a Hexagon; and so on."
This rule is not the case when dealing with isosceles triangles (Soldiers and Workmen), for their evolution occurs through
eventually achieving the status of an equilateral triangle, removing them from serfdom. The most acute angle of an isosceles
triangle gains thirty minutes (half a degree) each generation. Additionally, the rule does not seem to apply to many-sided
polygons; often the sons of several hundred-sided polygons will often develop fifty or more sides than their parents.
The book poses several interesting thoughts, including the idea that higher dimensional beings have god-like powers over lesser
dimensions. In the book, the three-dimensional Sphere has the ability to stand inches away from a Flatlander and observe them
without being seen, can remove flatland objects from closed containers and "teleport" them via the third dimension apparently
without traversing the space in between, and is capable of seeing and touching the inside and outside of everything in the two
dimensional universe; at one point, the Sphere gently pokes the narrator's intestines and launches him into three dimensions as
proof of his powers. The book implies the possibility that higher dimensions than three exist, implies a satirical description of
Victorian life, and teaches a lesson about ignorance, closed-mindedness, and self-satisfaction.
Number 74 Coordination Polyhedra
A simple site still under construction.
Number 75 Fractal Polyhedra
An interesting idea of models made of many polyhedra diminishing in size.
Number 76 Geometry Junkyard
An absolute must. There is so much on this site to satisfy all geometers. It is continuously being
Number 77 Arts, Mathematics and Architecture
A list of very good links. Each one of the links is worth a visit.
Number 78 Crystallographic Polyhedra
An interactive site illustrating 29 polyhedra in wireform in Applets.
Number 79 Sandor Karbai (Hungary)
This site has a lot of gifs that would make good classroom posters. Unfortunately they are not in
English but this is offset by the clarity of the illustrations. There are some excellent ‘colouring in’
books and some interesting lessons in Polyhedra.
Number 80 Korthals Altes Paper Models of Polyhedra
This site has so many nets for polyhedra. A site well worth visiting time and time again.
Number 81 Playing to Win
A statistics site that may be useful.
Number 82 Polyhedra by Peter Cromwell
This is a book review
Number 83 Origami Polyhedra
A very good site with very clear explanations of method. There are a lot of models to create using
this module technique.
Number 84 Euler’s Formula
This is a simple site with a few illustrations.
Number 85 Tetrahedral Puzzles
Dissection puzzles that would make good classroom activities.
Number 86 The Platonic Solids
Here is a good little treatise on the Platonics.
Number 87 - 88 Uniform Polyhedra 2 PDF polyhedra-my polyhedra.pdf
Here is a rather technical approach to the subject of Polyhedra.
Number 89 Polyhedral Links and Reviews
Some reviews worth reading.
Number 90 The Uniform Polyhedra
A well illustrated site covering the 80 Uniform Polyhedra. User friendly and well worth a visit.
Number 91 Puzzles
These puzzles are all available on a CD and cover a wide range of Mathematics.
Number 92 Mathsteacher
This is an interactive site for years 7 to 10. The software that is for sale may be worth considering.
Number 93 Basic Problems of Polyhedra
The basic problems of colouring Polyhedra and labyrinths on Polyhedra are well covered in this site
and there is also a set of about 100 problems of these types together with their answers to download
(zip format).
Number 94 Geodesic Dome Under the Paw of the Lion
Some interesting photos of a Geodesic sphere in an ancient sculpture.
Number 95 Weaving Paper Polyhedra
Step by step instructions to complete a number of Polyhedra are given. These instructions are clear
and easy to follow.
Number 96 - 97 The Beach Cleanup and the Whale
An interesting site on Statistics with a practical aspect.
Number 98 New Zealand Maths
Worth a look.
Number 99 Poly
Poly is shareware, which is helpful in making polyhedra. Available for PC and Mac.
Number 100 - 101 Polyedergarten
There are a great many attractive models on this site. There are no nets but there should be enough
information to inspire people to design for themselves and produce something unique. There is also a
list of links in Number 101
Number 102 Volumetric Relationships
Interesting site showing that there are some simple relationships between the volumes of some
Number 103 Book on Modular Origami
May be worth getting a copy of this book for the Maths Department or School library.
Number 104 Polyhedral Maps
An excellent site for creating “globes” on Polyhedra. A Good project for the class.
Number 105 Puzzles in Polyhedra
A challenging site for teachers, students and parents. Well worth a look.
Number 106 - 110 Guy Inchbald’s pages
These sites are an interesting group of pages with a fair amount theory and excellent diagrams.
Number 111 Jitterbug Defined Polyhedra
An interesting way of showing Polyhedra and their Duals. There is also a great deal of theory and
calculations in these pages.
Number112 Polyhedra Encyclopedia
An excellent site giving many Polyhedra with PDFs on the various models with their calculations.
Number 113 Triangles, Tessellations and Polyhedra
A good site on tessellations. There is also an interactive online tiler available.
Number 114 Mathematics Dictionary
An extensive interactive Mathematics dictionary. Very useful.
Number 115 Polyhedra Blender
Software that can be freely downloaded. May be worth playing with.
Number 116 Spidron System
I found the ideas in this site interesting, exciting and challenging. Some ideas here worth
experimenting with.
Number 117 - 118 Penrose Tilings
Good sites on tilings and Escher type patterns. Students would enjoy being creative.
Number 119 Ministry of Education on-line resources
Plenty here for everyone.
Number 120 Spherical Geometry
A great site on Spherical Geometry. Well illustrated, good theory and nets. Plenty to learn from this
Number 121 - 122 Steven Dutch’s site
This site shows the Archimedean Polyhedra and their tilings. Steven Dutch’s site is a very detailed,
many paged, site that has so much information on tessellations and polyheda that it is mind boggling.
A must for serious polyhedrists.
Number 123 The Icosahedron and the Golden Section
The illustrations on this site show very simply the relationship between the Golden Section and both
the Icosahedron and the Dodecahedron (Duals of each other.)
Number 124 Jo Malkevitch
This short page gives a potted view of the connection between Perspective, Architecture and Nets.
Number 125 - 126 Krystina Burczuk’s Origamic Gallery
These sites have a huge amount of Polyhedra made using the module technique. This method lends
itself to group work.
Number 127 Uniform Polyhedra
A number of pages of very good diagrams that can be used for posters etc. Just illustrations. No Nets.
No calculations.
Number 128 General Studies
A web site worth looking in as it covers a huge range of material not just Mathematics.
Number 129 Pythagoras’ Theorem
An animated showing of Pythagoras. Interesting proof.
Number 130 ErithmAttack
Free downloadable software on Arithmetic. A good resource on quick testing in a fun way. The
students should enjoy this as they are not directly in competition with other students.
Number 131 I Know That
A site for primers to Standard 6. This is a good site.
Number 132 Cyber Challenge
An excellent site with a timed set of questions testing the basic operations. Great for kids.
Number 133 – 136 Mathsnet
This site is EXCELLENT. It covers the curriculum and has many useful test questions.
Number 137 -140 nrich maths
Quite a few puzzles of varying types, which could be challenging for your students. There is a
weekly problem and there are nrich books and CDs to purchase. I think it is worth a visit.
Number 141 Mathemagician
Another interactive site dealing with basic number operations. Could be fun for lower ability
Number 142 Names of the powers of 10
A handy reference for powers.
Number 143 NCTM
This is an interactive site that has many facets from algebra through to probability & statistics. The
site would be well worth exploring and could become addictive.
Number 144 That Quiz
There is a lot to do and the tests give opportunity to correct errors. The following quote sums up the
essence of this site…
“The best online resource for mathematics on the web today with over 400,000 graded exams to date and
over 50,000 participating students. What we believe in: Clean, quality, easily accessible, educational software
for every school and child regardless of geographic location or economic class. What we don't believe in:
Games, advertising, fees, spam or gimmicks. ThatQuiz is free for educational use.”
Number 145 Multple Choice Tests
These tests have muliple choice answers and are challenging to students. Well worth a visit as the
page can be printed as a worksheet. Check out the Interactive tests – ‘Table of Contents’ at the
bottom of the page. There is also an extensive amount of Maths links.
Number 146 OnLine Maths Library
This OnLine Library is totally free including the software required to access it. The following quote
from the site should explain the objectives of the site…
“The Mathwright Library is our gallery. It is a growing collection of interactive, electronic
mathematics and science books that range in size from 1 to 55 pages (or more). Our
collection presently holds roughly 250 books, that comprise 1800 interactive pages. Each
page is like an Applet.”
Number 147 Got a Problem?
This page only deals with fractions but if you click on Home Page you will find that there is much
more that you can get help with. Just go to the appropriate section and type in your question.
Number 148 The Golden Section in Architecture
A very interesting site that branches in many directions. The site covers or leads to art, music,
fibonacci and much more. You must take a look.
Number 149 A lesson on Fibonacci
A lesson from a Homeschool website.
Number 150 Golden Section
An interesting site to view.
Number 151 -152 The Pentagram and the Golden Section
Another very interesting presentation of the Golden Section that is well illustrated and worth visiting.
Number 153 Pythagoras Theorem
This is a good site on Pythagoras.
Number 154 What does it mean to be Golden?
Good descriptions, links and lesson-plans to use.
Number 155 Softeware for the Golden Section
Free Software on the Golden Section. I have not personally downloaded it but it might be worth
looking at.
Number156 Cartoons
No list of websites is complete without one website on cartoons.
Number 157 Inca Mathematics
Inca Mathematics and much more. There are proofs of many Geometric Theorems & Problems, interactive
jigsaw type puzzles, quizzes, quotes, art and poetry. The Triangle Centre covers just about everything that
you need to know about triangles. An Excellent site to visit.
Number158 Wikipedia definition of the Golden Section
I recommend Wikipedia as a source of information.
Number 159 Wikipedia on Gradians
This is the only reference that I have found on Gradians. I have often wondered why it is on our calculators.
Number 160 Wikipedia on Polyhedra
Number161 Zero Directory
Yet another excellent website with a lot of interesting information.
Number 162 Calendars
A very extensive site on all sorts of Calendars. Could be useful.
Number 163 Practical Formulae
A wide range of very useful formulae. (Even automotive formulae)
Number 164 Games
A series of games for two players. The games are explained and theory on strategies is given.
Number165 Humour
A good mathematical joke is better than a dozen mediocre papers.
Number 166 Numericana’s Index
This is a must site. There is a huge variety of interesting information on this site together with more links.
Number 167 Magic
Some simple maths tricks and also a link to another site.
Number 168 Roman Numerals
All you need to know about Roman Numerals.
Number 169 Prefixes
This is the ultimate list in prefixes.
Number 170 The Isohedra
There are 30 Isohedra each with an even number of sides and all of which make fair dice.
Number 171 Diversions in Maths and Numbers
Some book reviews plus a lot of other information and trivia.
Number I72 Mike Keith’s Poly
Downloadable software.
Number 173 Mike Keith’s Poly
Downloadable software Zip
Number 174 - 175 Movie of the book 59 Icosahedra
This is an interesting look at the 59 Icosahedra.
Number 176 Trig Without Tears
An excellent site for Trigonometry. Well worth a visit.
Number 177 Buckmaster Fuller Institute
An overview of the work of Buckmaster Fuller.
Number 178 Games, Puzzles, Illusions and Sudoko
Really good, entertaining site with so much to do and so many ideas to glean for the classroom.
There many hundreds of these puzzles etc together with their solutions and the puzzzles can be printed off.
Number 179 Brainteasers
Quote from the site ……..
‘With over 10,000 brain teasers, riddles, logic problems and mind puzzles submitted and ranked by
users like you, Braingle has the largest collection anywhere on the internet. Our large array of unique online multiplayer games will
keep you entertained for hours, and if you want to improve your brain, check out the Mentalrobics section. If you crave a brain teaser,
mind puzzle, riddle or game, we are the place to get it. Increase your creativity, boost learning and become a better person at Braingle.
Get ready to have your brain tangled!’
Number 180 Tetrahedral Kite using Straws
This site gives full instructions on how to build and fly this kite. Maybe a class activity working in pairs and in
competition with each other.
Number 181 Fibonacci Facts
With a brief introduction to Fibonacci the site gives a very good bibliography on the following areas
where Fibonacci is involved
1. BOTANY, BIOLOGY. The growth patterns of plants; the geneological tree of the male bee; the
crossroads of mathematics and biology
2. PHYSICAL SCIENCES. Atomic fission asymmetries; ladder and cascade electronic network
analysis; reflection paths of light; musical properties; computer programming and search strategies;
tributary patterns of streams and drainage patterns; fractal branching of diffusion aggregates.
3. BUSINESS AND ECONOMICS, EDUCATION, POETRY. Stock market cycles; business
cycles; teaching slow learners; analyzing poetry.
ARTS. The Great Pyramid of Gizeh; harmonic design in Minoan architecture; the Parthenon of the
Acropolis in Athens; ancient Roman mosaics; the Golden Ratio in art and architecture.
Number 182 Magnus Wenninger
This site is loaded with photographs of Wenninger’s models and is a great source of inspiration.
Number 183 Jaap’s Puzzle Page
If you are into moveable puzzles then this is the site for you. There is an amazing amount of this type of
puzzle with solutions. Many of them can be done on screen using Java Applets. There is also a list of Maths
related articles that look very interesting.
Number 184 Geodesics
This is a commercial site but this part of the site is well worth investigating.
Number 185 Regular Polyhedra
The 5 basic Polyhedra together with some calculations.
Number 186 Pavilion of Polyhedreality
Any site with George Hart in it is worth visiting. There are certainly some surprising constructions on this site.
Number 187 Uniform Polytopes in Four Dimensions
The following is a statement from the website…
“This is the world’s only website that tabulates all the convex uniform (i.e., Platonic and
Archimedean) polychora (that is, four-dimensional polytopes), and until Norman W.
Johnson’s book Uniform Polytopes is published by Cambridge University Press, it remains
the only place in the world where you can find this information!”
It is certainly interesting and has information plus.
Number 188 Encyclopedia of Polyhedra
As I have stated before George Hart is well worth a visit and this site is no exception. You can spend a lot of
time here and learn a great deal of information.
Number 189 The Geometry Junkyard
This site ia a favourite of mine that I continually go back to. The following quote will explain the site’s
philosophy...“These pages contain use net clippings, web pointers, lecture notes, research excerpts,
papers, abstracts, programs, problems, and other stuff related to discrete and computational
geometry. Some of it is quite serious, but I hope much of it is also entertaining. The main criteria for
adding something here are that it be geometrical (obviously) and that it not fit into my other
geometry page, Geometry in Action, which is more devoted to applications and less to pure math. I
also have another page on non-geometrical recreational math.”
Number 190 - 191 Platonic and Archimedean Polyhedra
I intend using the fold up patterns on this site as posters in my classroom
The second site is for the book Sacred Design Geometry Sourcebook. Would be a good addition to the Maths
Number 192 More Puzzles
Yet another good source of activities with plenty of Geometry puzzles.
Number 193 Mahobe Books
A New Zealand website for Geometry Maths books.
Number 194 Kaleido Software
Software to download. Interesting page.
Number 195 Sliding Triangle Puzzle
This interactive puzzle has various levels of difficulty and could be a challenge to students’ skills in logic.
Number 196 All Uniform Polyhedra
A site with just pictures and basic information. Unfortunately there are no nets of the Uniform Polyhedra.
Number 197 Archimedes
A great site for studying this great Mathematician.
Number 198 The Golden Section Ratio ---- Phi
An excellent site on Phi.
Number 199 Golden Section
An interesting site but be warned … quote from the site… “WARNING: BORING MATH AHEAD!
Number 200 Cone Generator Software
Software for cone generation. Could be useful.
Number 201
This is a very interesting site (rather technical), particularly the second address that has a great many
diagrams that can be downloaded as posters. Just right click on the image and save into whereever.
Number 202 Triangular Geodesics
For those of you that want to get down to the nitty gritty of Geodesics.
Number 203 The Geometry Centre
A good site for the students with plenty for them to discover and maybe discus. See the activities
page. Information on tilings and tessellations is done in reasonable depth. There is also a great
tetrahedral puzzle to make. There are several very good interactive patterning progammes as well.
Number 204 Uniform Polyhedra
Just good illustrations. There are some good links.
Number 205 -206 Guy’s Polyhedra Pages
A fascinating site. A few hours spent here will be well worthwhile. Great illustrations.
Number 207 Problem of the Month Archive
A site with regular problems to challenge the minds of interested people.
Number 208 Magic Cube 4D
This site is for Rubik’s Cube fans.
Number 209 Polyhedra Nets
10 Basic Nets to print off.
Number 210 Polyhedra
New Zealand website listing good links to Polyhedra Sites.
Number 211 Ministry of Education New Zealand
Ministry sites plus other links.
Number 212 Enumeration of Polyhedra
A site that is a little out of the ordinary and not really for the average student.
Number 213 A Site for Kids
A fun site that has many branches. There is something for everyone and it is worth a visit.
Number 214 Polyhedra Nets
There are some good nets available here.
Number 215 A Great Collection of Nets
More than enough nets to keep a class going for a long time.
Number 216 Maths Forum
A site where students can seek answers to their maths problems.
Number 217 Mathworld
There is so much on this site that it has to go in the top ten sites
Number 218 Poly Software
Poly is a shareware program for exploring and constructing polyhedra. With Poly, you can
manipulate polyhedral solids on the computer in a variety of ways. Flattened versions (nets) of
polyhedra may be printed and then cut out, folded, and taped or glued, to produce three-dimensional
models. Poly is used in schools and homes around the world.
Number 219 An online encyclopaedia of polyhedra and related topics
Worth a peek.
Number 220 The Phinest source of the Golden Section (this is not a spelling mistake)
This is an excellent site covering a vast area such as the Golden Section in Life
The following is a list of accessible sites of specific interest. Try some of them.
Human Hand
Human Face
Human Body
Human Beauty
Human Teeth
Human Heartbeat
Human Health
Body Temperatures
Animals 1
Animals 2
Population Growth
Bucky Balls
Phi in Circles
Penrose Tiling
Phi Formula Geometry
Phi Mandalas
Here is another list of Websites that you may like to visit. (Some of the sites
may have already been included above.)
My sculpture gallery
My annotated bibliography of polyhedra references
Zometool --- my favorite polyhedron construction set (my zometool page)
Jill Britton's page of polyhedra links
Magnus Wenninger's home page
Ulrich Mikloweit's incredible paper models
Vladimir Bulatov's raytraced and VRML polyhedra
Kris Manske's polyhedra pages
Roman Maeder's uniform polyhedra Pages
Tom Gettys' polyhedra Pages
Ha Quang Le's VRML polyhedra and Maple code
Mike Rollins' Polyhedra in Light Bulbs
Junichi Yananose's puzzles and VRML constructions
Jim McNeill's VRML polyhedra
Henry Chasey's polyhedra model collection
Jan de Koning's wooden polyhedra models
Bob Allanson's polyhedra pages and polyhedron generator
Scott Vorthman's Zomeworks software
Edwin Otoole's ring kits
Rolf Asmund's polyhedra pages
Steve Dutch's polyhedra pages
Tom York's polyhedron program
Daniel B. Grunberg's Poly program
Izidor Hafner's visualization olympiad and rhombic polyhedra
Mike Eisenberg and Ann Nishioka's HyperGami
George Olshevsky's paper models and polytope page
Polydron --- a snap-together polygon construction set
Polymorf --- another, but which allows more than two faces per edge
Peter Cromwell's Polyhedra book
Tom Lechner's polyhedra
Anthony Thyssen's giant polyhedral kites
Stewart T. Coffin's Puzzling World of Polyhedral Dissections
Greg Egin's polyhedral (and other) applets
VRML crystal structures at the Institut Laue - Langevin
dForm's VRML constructions of Platonic polyhedra
Farideh Dormishian and Rudy Rucker's Hypercube 98
Lexington High School's World's Largest Icosahedron
Kenneth Snelson's tensegrity structures
Leigh Boileau's wire models
Tom Longtin's Polyhedral (and other) images
Mark's Newbold's stereoscopic animated hyperspace objects
Pan Dragon's polyhedral animations
Daniel Green's geometry pages
Sam Gratrix's ray-traced polyhedra
Robert Dawson's ray-traced polyhedra
Mathematica Graphics Gallery
Stewart Dickson's Polyhedral Constructions
Steffan Scheller's polyhedral stuff
Doug Zongker's Polyhedral templates
Bathsheba Grossman's geometric sculptures
David Eppstein's Geometry Junkyard
Stephan Werbeck's dodecahedron constructions
Dave Rusin's notes on polyhedra
Gordon Kindlmann's peek at polytope slices
Laurens Lapré's polyhedra images
Martin Trump's random polyhedra pictures
Philo's Polyhedral Panoramic Photos
Rona Gurkewitz's Modular Origami Polyhedra
Jim Plank's Origami Page
Tom Hull's Modular Origami
Krystyna Burczyk's Modular Origami
Meenakshi's Modular Origami
Matt Holland's modular origami
John Sharp's Sliceform models
Jack Snoeyink's Un-disassemble-able Object.
Kirby Urner's Synergetics Home Page.
Steve Waterman's polyhedra with lattice-point vertices
The group for hyperspace
The Geometry Center
The Geometry Forum
Linda Stannard's Principles of Virus Structure
Icosahedral Virus Images (U. Wisconsin)
Buckminster Fuller FAQ
Paul Flavin's Polyhedra viewer
Richard Hawkins' Synergetic Geometry
Bill Collins's Tensegrity Structure Gallery
Uxbridge High School's stained glass truncated icosahedron
2000 polyhedra at Wallsend Central Middle School
Vedder Wright's Forksphere
NetLib database
Zvi Har'El's Kaleido Program
Good Construction advice (J. Conway)
Glitter Globe (pentakis dodecahedron circuit kit)
Section B Flag Construction
construction sheet
algeria | asturias (spain) | bahrain | benin - construction sheet | bosnia and herzegovina - construction
sheet | british columbia: construction sheet | burundi flag construction sheet | civil ensign (israel) |
construction details of the australian flag | construction sheet and flag variants (jordan) | construction
sheet: bangladesh | district flags (berlin, germany) | european union: official specifications for the
flag | french polynesia | georgia: national flag, 1990-2004 | georgia: rejected flag proposal, 2004 |
georgia | kuwait | macedonia: the "sun of vergina" flag (1992-1995) | macedonia | madrid community
(spain) | marshall islands | mexico | nauru | nepal | philippines - flag specifications | podgorica
(municipality, montenegro) | south african flag observed more closely | specifications and
construction sheet (andorra) | specifications and construction sheet (israel) | specifications and
construction sheet (malaysia) | specifications and construction sheet (malta) | specifications and
construction sheet (united arab emirates) | specifications and construction sheets (oman) | state flag
construction sheets (mecklenburg-west pomerania, germany) | sudanese flag law and contruction
sheet | swastika flag specifications and construction sheet (germany) | trbovlje (municipality,
slovenia) | turkey | turkish republic of northern cyprus | vojvodina (autonomous province, serbia) |
cape verde | construction details of the soviet flags | nepali flag law and construction details |
Section C Origamic Architecture
Section D Paper Models
Section E Bibliography
Mathematical Models
Cundy and Rollett
Paper Folding for the Mathematics Class
Donavan A. Johnson
Polyhedron Models
Magnus J. Wenninger
Spherical Models
Magnus J. Wenninger
Polyhedra Models for the Classroom
Magnus J. Wenninger
Patterns in Space
Col. R.S.Beard
Creative Publications
Mathematical Curiosities 1
Jenkins & Wild
Tarquin Publications
Mathematical Curiosities 2
Jenkins & Wild
Tarquin Publications
Mathematical Curiosities 3
Jenkins & Wild
Tarquin Publications
Polyhedra Primer
P Pearce & S Pearce
Dale Seymour
Designs From Mathematical Patterns
Bezuska, Kenney & Silvey
Creative Publications
Regular Polytopes
Vexilollogy is the study of Flags. It is another hobby of mine that has helped me to broaden
my knowledge of the History, Geography and cultures of the countries of the world.
The following list of websites covers an incredible amount of information on Flags, Emblems
and Heraldry. These sites are alphabetically listed rather than in any particular priority but within this
set of websites there is Art, Symmetry, Geography and History and I am sure that visits to the sites
would not be wasted time. Some I have put **** as sites that I like and have used a fair amount.
80 Puerto Rico flags and Coats of Arms. Site is in Spanish. ****
Les emblèmes de France.
Basque Country. Many flags
This is Flags of theWorld. ****
Very good French Interactive site. General information and maps.
Flags of the Russian Federation.
Taiwanese site. Go to Sub Web. Excellent portrayal of the history
of the flags of all Nations.
The Low Countries. Huge amount of flags.
Francophonie flags & French Flags
Great Geography site. There are flags.
Polish flags and history. (in Polish)
Military flags from all over.
Flags of the Native Peoples of the United States.
Czech site. World coverage.
Dutch website.
World of Flags
Over 1000 Czech flags.
online collection of flag photos
Administative divisions of the World. There is just data no flags as such.
Geographique Mondial. Flags and Photos of the countries
Flags of Swiss communes.
Italian flags.
European Military Flags 1700-1914.
Portuguese flags and maps.
In English and very good.
In English & German. An informative site.
German site for finding flags heraldry and location.
Flags and a quiz.
World Data Base. Excellent.
Russian site. Flags set in various groups.
Commercial but has good detail.
History, Geography, flags, anthems etc.
Flag reports in the news.
Recreational Boating.
Mongolian Flags and Arms.
Interesting site with lots of little extras.
Russian site with both heraldry and flags. A puzzle to find the info.
Excellent Heraldry site
All the flags on 8 pages
Danish commercial site but well worth a visit.
Chinese site
North American Vexilollogical Association.
Largest website on International Heraldry. Very Good
World flags from a French site.
A World of Flags!
There are 1692 images in the database, today 18 Sep 2006
Excellent. ****
Great Italian flag site that has some interesting picture cards, (cigarette cards maybe), portraying
flags. ****
Native American Flags.
Spanish Flags and Coats of Arms on a Spanish site. ****
Flags, Geography and History. It’s all in French.
German site
Russian site with some Englsh content. Plenty of Russian flags from all aspects of life. ****
Spanish Vexillilogical society
A site that is colour coded according to continent. All easily accessed. ****
Many flags and a great deal of historical background. ****
French text. Flags and info. Pictures and maps of the countries.
Heraldry. A site with many Coats of Arms. Blazons of Russian towns.
In French. A very good and informative site. ****
German Flag Society. Some parts of the site are in English.
This Site is in French and covers Frech Flags and Coats of arms. ****
Interesting Italian site which is reasonably easy to visit and get around.
Ties in with No 54.
Russian site – no English but an interesting site.
A great site in both English and Hungarian. The flags and arms are all Hungarian. ****
Upper Bavaria.
In at least 5 languages including English. Interactive site. ****
A site that helps to identify flags. ****
Flag Quiz.
Flag Identifier. ****
An English vexilloligical site.
Taiwanese site with flags but no history or information. Easy access.
Pictures of Native American Flags.
Australian site that gives the Flag Days for the Nations.