Thanks, Common Core. Physics blogger Chad Orzel writes about the way kids do math now. (Spoiler: he likes it.) [more inside]
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]
Alexander Grothendieck, who brought much of contemporary mathematics into being with the force of his uncompromising vision, is dead at 86, some twenty-five years after leaving academic mathematics and retreating into a spiritual seclusion in the countryside. "As if summoned from the void," a two-part account of Grothendieck's life, from the Notices of the American Math Society: part I, part II. [more inside]
Do you have fond memories of afternoons spent making geometric patterns and designs with your Spirograph? If so, enjoy Inspirograph, a digital version. [more inside]
Bees & Bombs is a tumblr of hypnotic GIF animations programmed by Dublin-based physics student Dave Whyte
The Foehr Reef is part of the worldwide Crochet Coral Reef Project. It was made by over 700 women and combines more than 4000 individual pieces of marine wonder. A short video shows its beauty [alternating English and German audio]. PDFs with pictures. "The Crochet Coral Reef is a woolly celebration of the intersection of higher geometry and feminine handicraft, and a testimony to the disappearing wonders of the marine world." It originated out of a desire to increase awareness of environmental threats to the world's reefs and is a conjunction of art, environmentalism, and geometry. [more inside]
Trio for Three Angles (1968) is one of many beautiful acclaimed visually-oriented short films with music by mathematical filmmakers Bruce and Katharine Cornwell, some animated by hand and some using early digital technology. It inspired three sequels: Similar Triangles (1975), Congruent Triangles (1976), and Journey to the Center of a Triangle (1978) (previously). [more inside]
Polyhedra and the Media - On the new polyhedra of Schein and Gayed, and mathematical journalism.
Geogebra is an interactive geometry tool which started as a free clone of Geometer's Sketchpad, but is now also an algebra, statistics and calculus tool. It is available for download for Windows, Mac, Linux, iOS and Android, or as a web app. [more inside]
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.
A standard 6 sided die is a cube. It has eleven nets. The sum of the numbers on opposite faces of a die is 7. [more inside]
Paper Matrix is a blog that gives instructions for cool papercraft objects, "reinterpreting the Danish tradition of woven paper hearts and ornaments." Cut paper in the prescribed ways and weave it together carefully to make a mobile of colorful hot air balloons, gorgeous and complex boxes; simple but satisfying pennants and much more... including a full theater for performances by paper dolls.
The regular polygons have been kidnapped by ninjas. Are you a bad enough dude/tte to construct the regular polygons with nothing but a virtual compass and straightedge? [more inside]
People have long been interested in the architectural endeavors of animals. The internal structure of bee hives, the hexagonal combs of wax, have been amongst these ponderings, going back to Marcus Terentius Varro's Rerum Rusticarum Libri Tres, a volume on Roman farm management. He wrote, "The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space," and over the years, mathematicians have studied the hexagonal structures made by bees, and in 1998, Thomas Hales produced a mathematical proof for the classical hexagonal honeycomb conjecture, which "asserts that the most efficient partition of the plane into equal areas is the regular hexagonal tiling." [more inside]
Mathematicians Henry Segerman and Saul Schleimer have produced a triple gear, three linked gears in space that can rotate together. A short writeup of the topology and geometry behind the triple gear on the arXiv.
Emma Kunz was a telepathic healer and researcher. Even though she didn't consider herself an artist, the hypnotic symmetry found in her hand-drawn healing charts is breathtaking.
The Royal Spanish Library has put online today an interactive version of Leonardo da Vinci's Madrid Codices I & II. There are transcriptions of the text (in Spanish and Italian, click "T" on the bottom menu), animations of many of the mechanical contraptions (click play button "ver animacion") and the "Indice" in the bottom menu organizes the folios by theme.
And collapsible hexagons are, I suppose, cool enough to at least amuse you a little bit during your class...
To keep yourself amused during your math class, you start playing with [all these strips of paper]. And by you, I mean Arthur H. Stone in 1939. (SLYT)
"The real satisfaction from mathematics is in learning from others and sharing with others." William Thurston, one of the greatest mathematicians of the 20th century, has died. He revolutionized topology and geometry, insisting always that geometric intuition and understanding played just as important a role in mathematical discovery as did the austere formalism championed by the school of Grothendieck. Thurston's views on the relation between mathematical understanding and formal proof are summed up in his essay "On Proof and Progress in Mathematics." [more inside]
A large selection geometric shapes, all created with just one HTML element & CSS. An interesting take on geometry. Includes Space Invader. [more inside]
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.
The best known packings of equal circles within a circle. Best packings with 1-12 circles. Best packings with 49-60 circles. Best packings with 1093-1104 circles. Also, circles whose areas form a harmonic series. Circles in an isosceles right triangle. Or generate your own circle packings. (Background for beginners: circle packing. Background for experts: circle packing.)
Henry Segerman creates mathematical sculptures using 3D printing: Round Möbius Strip, Hopf Fibration, Half of a 120-cell, Rectified Tesseract, Tesseract and 16-cell, Hilbert Curve, Knotted Cogs, Round Klein Bottle [more inside]
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]
Snowdecahedron. When life hands you a blizzard, make a Platonic solid. "Temporary public art" from Dan Sternof Beyer.
Measure-theoretic probability: Why it should be learnt and how to get started. The clickable chart of distribution relationships. Just two of the interesting and informative probability resources I've learned about, along with countless other tidbits of information, from statistician John D. Cook's blog and his probability fact-of-the-day Twitter feed ProbFact. John also has daily tip and fact Twitter feeds for Windows keyboard shortcuts, regular expressions, TeX and LaTeX, algebra and number theory, topology and geometry, real and complex analysis, and beginning tomorrow, computer science and statistics.
Let's say you're me and you're in math class, and you're supposed to be learning about factoring. Trouble is, your teacher is too busy trying to convince you that factoring is a useful skill for the average person to know with real-world applications ranging from passing your state exams all the way to getting a higher SAT score and unfortunately does not have the time to show you why factoring is actually interesting. It's perfectly reasonable for you to get bored in this situation. So like any reasonable person, you start doodling.[more inside]
Geometry, Surfaces, Curves, Polyhedra (many of which are beautiful) l Google Earth Fractals l fractals and chaos. [more inside]
When he was 32, his life seemed hopeless. He was bankrupt and without a job. He was grief stricken over the death of his first child and he had a wife and a newborn to support. Drinking heavily, he contemplated suicide. Instead, he decided decided that his life was not his to throw away: it belonged to the universe. Buckminster Fuller embarked on "an experiment to discover what the little, penniless, unknown individual might be able to do effectively on behalf of all humanity." If the architect, author, designer, inventor, and futurist Richard Buckminster "Bucky" Fuller were still alive, he would be 115 years old today. Though he died in 1983, his legacy grows on through recordings of his ideas and the Buckminster Fuller Institute. [more inside]
There's always been hyperbole in fashion; but fashion became truly hyperbolic this week when mathematican William Thurston, winner of a 1982 Fields Medal for his revolutionary re-envisioning of low-dimensional topology and geometry, teamed up with designer Dai Fujiwara (of the house of Issey Miyake) to produce a Paris runway show based on the fundamental geometries of 3-dimensional spaces. Thurston and Fujiwara briefly interviewed. Thurston's famous essay "Proof and Progress in Mathematics" concerns, among other things, Thurston's belief that the production of mathematical understanding can be carried out by means other than the writing down of formal proofs (though fashion shows are not specifically mentioned.) Previously in wearable non-Euclidean geometry: Daina Taimina's hyperbolic skirt.
The eyeballing game: compare your best attempts at several instinctive everyday tasks - determining a point of convergence, bisecting an angle, finding the midpoint of a line - against mathematical certainty. In a more financial mood? Play Chartgame: given a random historical stock chart of an unnamed S&P 500 company, choose to buy and sell as time advances to see if you can beat the market.
For math geeks. How to Draw the Voronoi Diagram. Voronoi diagrams, as a geometric model are fascinating because they can be used to describe almost literally everything: from cell phone networks to radiolaria, at every scale: from quantum foam to cosmic foam. See also the Wallpaper Group: there are only 17 ways to fill a plane with a regular 2 dimensional pattern. Fred Scharmen [weblog home] is known as 765 and also produces a number of shapes, textures and patterns.
Pattern in Islamic Art - thousands of high quality, free pictures of various motifs, patterns and architectural elements of mosques and other structures from Asia to West Africa. [more inside]
Topology and Geometry Software by Jeff Weeks.
Jared Tarbell is a computer artist whose Gallery of Computation has been previously featured on Metafilter . Several years ago he began working with the Epilog Mini 24 laser cutter, cutting out flat cardboard pieces and assembling them into complex geometric shapes. His Flickr set “lased” documents his work. Recently he made the transition to a more traditional artistic medium; oiled walnut , for his stunning piece 2222 holes.
Structure Synth is an application for creating 3D structures from a set of user specified rules. It is an attempt to make a 3D version of Context Free.
Curtis Steiner is a Seattle businessman and artist who operates a local gift shop. Both his home and his shop have garnered positive press, but his greatest artistic achievement may be his piece entitled 1,000 blocks, which explores the permutations of the six facets of the cube.
Dave Bollinger is a computer artist that specializes in geometry. He creates both still images and short videos. Some videos are silent, like this unusual Pac-Man homage, and some have soundtracks. Some are in black and white and some are in color. His Flickr photostream categorizes still images by style. His current fascination seems to be with cubes and cubic lattices.
Mathematicians create videos that help in visualizing four-dimensional objects. Science News writes about it: seeing in four dimensions.
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