Imperfect Congruence - It is a curious fact that no edge-to-edge regular polygon tiling of the plane can include a pentagon ... This website explains the basic mathematics of a particular class of tilings of the plane, those involving regular polygons such as triangles or hexagons. As will be shown, certain combinations of regular polygons cannot be extended to a full tiling of the plane without involving additional shapes, such as rhombs. The site contains some commentary on Renaissance research on this subject carried out by two renowned figures, the mathematician-astronomer Johannes Kepler and the artist Albrecht Dürer. [more inside]
Two enjoyable chapters [PDF, 33 pages] from the book Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers. "This book does not purport to show you how to create precocious high achievers. It is just one person's story about things he tried with a half-dozen young children."
Pencil and Paper Games is devoted to games you can play with nothing more than a pencil and a piece of paper (some of which can be played on the site, for those who do not have access to a pencil and paper, or remember what those are.) [more inside]
In Honor of the Centennial of Martin Gardner's birth (October 21, 1914), we've lined up Thirty-One Tricks and Treats for you: Magazine articles, new and classic puzzles, unique video interviews, and lots more. ✤ The Nature of Things / Martin Gardner [46min video] ✤ The College Mathematics Journal, January 2012 dedicated to Gardner with all articles readable online.
The Peg Solitaire Army is a problem spun off from a classic recreation, and yet another example of the golden ratio turning up where you least expect it. If you want to look at the game more deeply, George Bell's solitaire pages are the ne plus ultra: There's more about the solitaire army (and variants), ... [more inside]
Welcome to Al Zimmermann's Programming Contests. You've entered an arena where demented computer programmers compete for glory and for some cool prizes. The current challenge is just about to come to an end, but you can peruse the previous contests and prepare for the new one starting next month.
Polyhedra and the Media - On the new polyhedra of Schein and Gayed, and mathematical journalism.
The Teaching of Arithmetic: The Story of an experiment. In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite - my new Three R's. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language. I picked out five rooms - three third grades, one combining the third and fourth grades, and one fifth grade. I asked the teachers if they would be willing to try the experiment.
Discovering Free Will (Part II, Part III) - a nice discussion of the Conway-Kochen "Free Will Theorem". [more inside]
Reaction-diffusion reactions used to design housewares, puzzles, and more. If you want to experiment yourself, you might get some ideas from the demos at WebGL Playground or you might use this brief intro as a jumping-off point.
Visual Patterns. Here are the first few steps. What's the equation?
The Hierarchy of Hexagons. School geometry seems to me one of the most lifeless topics in all of mathematics. And the worst of all? The hierarchy of quadrilaterals.
It's Saturday; why not think about the pigeonhole principle? Here are problems and more problems and what you might call a problem with the principle itself as it is often stated.
The Angel Problem. The Angel and the Devil play a game on an infinite chess board...
NumberADay - Every working day, we post a number and offer a selection of that number’s properties.
Morpion Solitaire is a very simple pencil-and-paper, line-drawing game for which the best possible score is not known! New records are still being set.
Building a Computer 1: Numerals - recently my kids have been asking me about how computers work. I like to give in-depth answers to such questions, so we set out on a quest to understand how they work... Follow-up parts 2 3 4 5 6 7 8 9 10 11 12 13 14 15.
This short computer graphics animation presents the regular 120-cell: a four dimensional polytope composed of 120 dodecahedra and also known as the hyperdodecahedron or hecatonicosachoron. [more inside]
Beaded Polyhedra ❂ More beadwork (mathematical and otherwise) by Gwen Fisher ❂ Still more beadwork galleries at beAdinfinitum ❂ Three-dimensional finite point groups and the symmetry of beaded beads [pdf - some algebra, but lots of illustrations]
A thread full of proofs without words at MathOverflow and quite a lot more of them courtesy of Google Books.
The OEIS Movie is simply a slideshow of one thousand plots from the Online Encyclopedia of Integer Sequences, at two plots per second with sequence-generated music. [more inside]
The Geometry of the Snail Ball [pdf] - an interesting article (with some DIY advice at the end) about a toy shop curiosity you may have encountered.
Plus magazine has compiled all their articles on mathematics and the arts into one handy-dandy page full of highly enjoyable articles ranging from limericks and screeching violins to the restoration of frescoes.
Mathematics Illuminated is a set of thirteen surveys in varied topics in mathematics, nicely produced with video, text, and interactive Flash gadgets for each of the topics.
Trigonometric Delights. This book is neither a textbook of trigonometry—of which there are many—nor a comprehensive history of the subject, of which there is almost none. It is an attempt to present selected topics in trigonometry from a historic point of view and to show their relevance to other sciences. It grew out of my love affair with the subject, but also out of my frustration at the way it is being taught in our colleges.
The Sexaholics of Truthteller Planet - yes, it's one of those rotten logic problems, one of many that can be found at Tanya Khovanova’s Math Guide to the MIT Mystery Hunt.
Calculus of Averages - Newton and Archimedes did not possess this knowledge. No mathematics professor today can provide this knowledge and depth of understanding. Author John Gabriel maintains a blog, Friend of Wisdom, and contributes articles such as Are real numbers uncountable? to Google's Knol project.
A gathering of puzzles including many old chestnuts but also perhaps one or two you haven't met before.
Who is Alexander Grothendieck? [PDF] This lecture is concerned not with Grothendieck's mathematics but with his very unusual life on the fringes of human society. In particular, there is, on the one hand, the question of why at the age of forty-two Grothendieck first of all resigned his professorship at the Institut des Hautes Etudes Scientifiques (IHES); then withdrew from mathematics completely; and finally broke off all connections to his colleagues, students, acquaintances, friends, as well as his own family, to live as a hermit in an unknown place. On the other hand, one would like to know what has occupied this restless and creative spirit since his withdrawal from mathematics.
If you could use a great big free handbook of discrete math and algorithms, Jörg Arndt's fxtbook wants to be your friend. Plain text table of contents to whet your appetite.
Lightning calculator and "mathemagician" Art Benjamin goes through his paces in a 15 minute video.
Nowhere-neat tilings are actually pretty neat. We all know you can't "square the circle", but do you know the story of squaring the square? (And by the way, even if you can't construct π with a ruler and compass, you can come awfully close without too much work.)
Symmetry. Shakespeare. Islamic medicine. Creative writing challenges. Four podcast series from University of Warwick.
The Prime Game is not really much of a game, but it is a neat & little-known fact about the decimal representation of prime numbers.
Here are some beautifully rendered views of polytopes, and a few more. The rendering program, Jenn 3D, is free and downloadable, (OS X, Linux, Win) and includes some really dazzling fly-about and camera effects as well as tons of high-dimensional models to explore. There's also a mind-boggling possibility of playing Go on boards in projective space. Via the Math Paint blog, which leads to other interesting places...
SlitherLink - a little spatial-numerical puzzle. Here's a better exposition of the rules from the puzzle's inventors, and another collection of puzzles. Oh, and a little survey of other sneaky, snaky puzzles.
Bending a soccer ball - mathematically. Found via Ivars Peterson's short exposition on Braungardt and Kotschick's The Classification of Football Patterns [pdf, technical].
Sphere and circle arrangements, the Droste effect, and more: mathematical imagery by Jos Leys. The Droste effect article is informative, too.
The Sarong Theorem Archive is the premier online repository for pictures of mathematicans in sarongs proving theorems.
Project Euler is a running contest of programming challenges to hone your algorithm skills. "Each problem is designed according to a 'one-minute rule', which means that although it may take several hours to design a successful algorithm with more difficult problems, an efficient implementation will allow a solution to be obtained on a modestly powered computer in less than one minute."
The Spidron is an interesting geometric construction that seems to lend itself to folding, dissection, and space-filling in two and three dimensions.
Michael Hutchings' rope trick and Dylan Thurston's two-handed knot-drawing sk1llz. Did you need to kill some time practicing pointless skills today?
The universe in just two symbols. The rest, as they say, is details. No wonder the "Physics Establishment" is trying to keep this quiet. The author, having conquered the universe in general, tackles poetry, as well.
Minimal surfaces in 3D (red/green, or stereo pairs), with rotate and zoom. If you want to go beyond the eye-candy aspect, here's the obligatory Mathworld link, the classic intuitive explanation, and a raft of additional information. If you like eye-candy, don't miss the ray-traced minimal surfaces and these interesting, but non-minimal surfaces.