| Mathematics_Museum_(Japan) At Mathematics Museum (Japan) you would be surprised how interesting mathematics is. You will find exhibition rooms produced by Japanese researchers and educators. |
| Mathematik Individual pages on different topics in Mathematics. Examples : group theory, dynamical systems theory, geometry or number theory. |
| Mathematische_Basteleien Topics include Flexagon, Soma Cube, Pentominos, Cube-it, Rubik's Cube, Froebel's Star, Tangram, House of Santa Claus, Chronogram, Numeric Palindromes, Latticework of Letters. English/German. |
| Mathematrix Devoted to exploring the more entertaining (and generally lesser known) areas of mathematics. Can be enjoyed by anyone, from individuals with little or no math background to professors of the subject. |
| Mathmos Includes puzzles, jokes, quotations, poetry, and FAQs. |
| Maze_Classification_and_Algorithms A short description of mazes and how to create them. Definition of different mazetypes and their algorithms. |
| Mudd_Math_Fun_Facts An archive of interesting math facts for use in the classroom or just for fun. Browse by subject, difficulty, keywords, or try the "random" feature. Based at Harvey Mudd College. |
| Narcissistic_Numbers Those that are representable, in some way, by mathematically manipulating the digits of the numbers themselves. |
| The_Nine_Digits_Page Puzzles and problems connected with numbers using the digits 1-9. |
| Number_Recreations_by_Shyam_Sunder_Gupta Features interesting facts about different numbers. Includes favorite related links. |
| On_the_Puzzles_with_Polyhedra_and_Numbers This is an article on a set of didactical games edited by the Portuguese Mathematical Society (SPM). |
| One_Metaphor_Fits_All Explains Conway's audioactive decay that is generated by a particular kind of sequence. Includes illustrations and related resources. |
| Properties_of_Dice Polyhedral dice and their properties. |
| Recmath Includes pages on magic squares and polyomino patterns and contains related java applets. |
| Recreational_and_Educational_Computing A newsletter with programs, including optional supplemental PC disk. All back issues are available. Topics include puzzles and teasers, BASIC programming, letters, graphics, fractals, challenges, r |
| Recreational_Mathematics Links collected at CAMEL, the Canadian Mathematical Society website. |
| Recreational_Mathematics_(David_Eppstein) An extensive list of web resources for recreational math. |
| Recreational_Mathematics_Forum A forum for posting messages about math recreations. Hosted at Delphi. |
| Recreational_Mathematics_Topics By Steven Dutch. Symmetry, Crystals, Polyhedra and Tilings; Pythagorean triplets and other things about sums of powers; Geometry Classics. |
| Roman_Numerals Contains a introduction to Roman numerals including a translation of the digits used and a converter which can convert decimal to Roman numerals and vice versa. |
| Rubik\'s_Cube_Lecture_Notes Notes on the mathematics of the Rubik's cube. |
| SimonSingh_net Home of Simon Singh: author, journalist and TV producer, specialising in science and mathematics. Cryptography is one of his specialties, and his site has a lot of educational and fun content about c |
| Sir_Roger_Penrose A article about him and his interests and contributions to recreational mathematics. |
| Skytopia_-_Super_Magnet A colourful world built entirely using mathematical atoms and molecules. Pictures and animations demonstrate structures colliding and interacting. Animated GIF demonstrations. |
| The_Sound_of_Mathematics Algorithmic music determined by mathematics and by the musical preferences of a human. General MIDI files. |
| Spirocharts A windows application that creates mathematically precise spirograph drawings; savable as images. |
| Spirograph A java applet for creating Spirograph images. |
| Sportlab Combines high school and college math topics into sports applications. |
| Stunning_Friends_with_Math_Magic A collection of card tricks, number guessing games, paper and glue magic, and other math exercises. |
| Table_of_Numbers_Problem Given a m * n rectangle, place all numbers from 1 to mn that minimizes the sum of the products of rows and columns (both in Spanish and English). |
| Treasure_Troves___Book_List Book list including titles, authors, publishers, prices, page count and some have links to Amazon.com. |
| Wade_Edward_Philpott Profile and description of his mathematical games and puzzles. |
| Wade_Edward_Philpott_Collection A special collection at the University of Calgary Library. |
| Who_Can_Name_the_Bigger_Number? An essay by Scott Aaronson on the quest for ever-bigger numbers, from exponentials to Busy Beavers. |
| Australian_Map_Circle The Australian Map Circle (AMC) is a national group of map producers, users and curators, which acts as a medium of communication for all those interested in maps. |
| The_British_Cartographic_Society an association of individuals and organisations dedicated to exploring and developing the world of maps |
| International_Map_Trade_Association_(IMTA) IMTA members are the companies that create and sell the maps, atlases and map-related products you use. Find information on maps, geography and mapping technology. |
| MapHist_Discussion_Group MapHist, the Map History Discussion List, is an e-mail discussion group whose primary focus is historical maps, atlases, globes and other cartographic documents. |
| Mt__Diablo_Surveyors_Historical_Society Society of Land Surveyors interested in tne history of Mount Diablo, Northern California, and the establishment of the Mt. Diablo meridian and initial point. |
| North_American_Cartographic_Information_Society NACIS is a professional society dedicated to the advancement and cooperation of all cartographic disciplines. Members include practicing cartographers, educators, map librarians, and map collectors. |
OK, I admit it - these weren't constructed entirely without computer assistance. Usually I write some C code to generate whatever the shape is and figure out which cells in a grid made up of 1x1x1 LEGO bricks should be filled in. The code outputs this as an LDraw .DAT file, separated into construction steps adding one complete layer of the structure in each step. Then I use MLCad to view the .DAT file. I play around with the parameters and repeat until I have something that looks nice and which will probably be able to balance. But that's the easy part. Now I have to try to construct it out of actual LEGO bricks so that it actually holds together. Worse, I have to use the collection of bricks actually in my possession and not currently in use in some other construction [in case anyone's interested, I've convinced myself that this is, in general, an NP-hard problem. One of these days I'll write up the proof]. This usually takes far longer than the first stage - and a few times I've had to give up. Occasionally I cheat just a teeny bit and deviate from the output of my code in order to help the structure hold together. I used to be able to say that all the models shown on this page were constructed using only the 1xN and 2xN standard bricks you can find in the blue bulk tubs. No plates, no L-shapes. That was until I built the Costa surface. I just couldn't make it hold together with bricks alone, so I gave in and used a few small plates too. And after these, all I can say is... the sooner The LEGO Group starts selling bulk packs of 1x3 bricks the better :-) When I get round to it, I'll add something here on the actual mathematics. For the moment, here are some pictures. Marvel at the wasted hours.... Click on the thumbnails to see more (and larger) pictures 
Various sizes of Mobius strip. These were the first mathematical sculptures I attempted. The big one was about my fourth (and I had to cheat a little bit with the top two layers to make it hold together). It's 37 bricks high and balances on a footprint contained in a 5x9 rectangle. It is essentially one brick thick throughout. Click on the thumbnail for more pictures and an LDRAW .DAT file... 
A trefoil knot. A friend at work suggested this after seeing the Mobius bands. This was my first attempt. It is very flimsy (and I cheated in two places by modifying the collection of occupied cells output by my program). Still, it proved it was possible. Click on the thumbnail for more pictures 
A figure eight knot. I made this thicker than the trefoil knot in order to help make it rigid. Nonetheless, I think this is the most difficult single construction I have ever made out of LEGO. Those long sweeping curves, hanging unsupported in space... It's only when you get about 2/3 of the way up that you start to discover exactly which bits 1/3 of the way up aren't strong enough. And there are never enough 1x3 bricks... But I didn't cheat anywhere. I'm very pleased with this. The figure-eight knot has a nice tetrahedral skew-symmetry which the model illustrates quite well. Click on the thumbnail for more pictures and an LDRAW .DAT file... 
Enneper's minimal surface. The same friend(?) at work (thanks, Fred) said that he thought minimal surfaces would probably be too hard, so.... Click on the thumbnail for more pictures and an LDRAW .DAT file... 
And just to prove the point... The Catalan minimal surface Until I made the Klein bottle, this was the largest of all my mathematical LEGO models. Despite the sheets being only one brick thick, the construction is remarkably rigid. I think this probably has to do with the zero curvature of the surface. though I don't really understand what's going on. I am now officially almost out of blue bricks... Click on the thumbnail for more pictures and an LDRAW .DAT file... 
A Klein Bottle A couple of people asked about a Klein bottle, so here you are. I spent ages finding a nice parametrization of this, so I hope you like it. Click on the thumbnail for more pictures and an LDRAW .DAT file... 
A Cross-Cap Click on the thumbnail for more pictures and an LDRAW .DAT file... 
Another Cross-Cap I wasn't very happy with the first one, so here's a completely different rendering of the cross-cap in 3-dimensional space. Click on the thumbnail for more pictures and an LDRAW .DAT file... 
A Costa Surface This was a bit of a challenge, but I've wanted to do one of these for ages. Click on the thumbnail for more pictures and an LDRAW .DAT file... 
Scherk's First Surface Not especially challenging, but a nice example of a saddle point Click on the thumbnail for more pictures and an LDRAW .DAT file... 
Bour's Surface The most elaborate and difficult yet. It left me with exactly six unused 1x3 yellow pieces. Click on the thumbnail for more pictures and an LDRAW .DAT file... 
New! Boy's Surface Construction was a nightmare and I nearly ran out of 2x4 blue bricks... Click on the thumbnail for more pictures... 
New! Punctured Torus Currently my favourite. Click on the thumbnail for more pictures... The contents of this page are Copyright © A. Lipson 2000-2006 LEGO ® is a trademark of The LEGO Group, who have nothing to do with this or any of my other LEGO-related web pages. Back to ASL's home page Back to ASL's LEGO page FAQs and contact information 
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