Math Toys

Trammel of Archimedes

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Trammel of Archimedes: as the shuttles take turns completing their straight line journeys, the end of the crank arm traces an ellipse. Sometimes sold as a “do nothing machine” or “nothing grinder”, far from doing nothing this simple and crucially important mechanism demonstrates how rotational motion can be converted into translational oscillatory motion- such as how a piston can drive an engine’s crankshaft. This version was crafted from fine maple, cherry, and oak by artisan Neal Olsen. 

Ambiguous Object Illusion Mug

From Etsy: BUY NOW Squirkle Mug Ambiguous Object Illusion 

These type of objects were invented by mathematician Kokichi Sugihara, and you can buy his books here: 
From Amazon: BUY NOW Ambiguous Objects by Kokichi Sugihara 
The math and physics are described here in this technical journal article by Prof. Sugihara

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Ambiguous Object Illusion Mug: circle or a square? It’s all a matter of perspective and viewing angle. The complex shape allows for both to be perceived and is based on the work of mathematician Kokichi Sugihara of Meiji University in Japan, the inventor of this illusion and art form. 

The Random Walker

Galton Board version available here: 
From Amazon: BUY NOW Galton Board 

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The Random Walker: second model of two Galton Boards designed and produced by IFA.com- this version is made to demonstrate probability in investment returns of a global stock market portfolio relating to risk capacity. Slow motion reveals the erratic path of each steel ball (second half of video). The red graph shows the distribution of 592 monthly returns (mean =1%, SD=5%) representing data from 50 years of an IFA Index fund- here the random “walk” of 3000 steel balls falling through 12 levels of branching paths always produce a close match, and both distributions tend toward the famous bell curve distribution. A wonderfully designed modern version of the Galton Box invented by Sir Francis Galton(1894) to demonstrate the Central Limit Theorem - showing how random processes gather around the mean.


Spherical Dice

A must for any die/dice collectors: 
From Amazon: BUY NOW Spherical Dice 

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Click this link for other amazing dice featured on @physicsfun 

Spherical Dice: these fair six "sided" dice are hollow inside with a ball that weights each sphere such that one of the six values is always on top. When these dice are rolled (literally!) the internal weight lands in one of six cavities inside creating a low center of mass which aligns one of the numbers to the top. Another low center of mass toy! 

Curves of Constant Width

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From Maths Gear: BUY NOW Curves of Constant Width (set of 4) 

Click here for 3D Solids of Constant Width

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Curves of Constant Width:  regardless of the precise shape, any curve of constant width has a perimeter equal to pi times its width! These convex shapes will roll because like a circle they have the same diameter from one side to the other no matter their orientation. Here are two famous examples: the Reuleaux triangle (found in rotary engines) and a Reuleaux pentagon- note how the two straightedge rulers remain parallel as the shapes rotate between them, just as one would expect circles to behave! These physical representations of the special curves seen here are produced by Maths Gear (Matt Parker and Steve Mould).

Shadow Stereographic Projection

These mathematical art objects are created by Henry Segerman and available here: 
From Shapeways: BUY NOW Mathematical Art 

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Wikipedia has a nice introduction to the math and applications of stereographic projection

Shadow Stereographic Projection: 3D printed sculptures that cast geometric shadows. When illuminated by a point source of light (placed at the top pole of the sphere) the shadow cast by the rays of light represent a one to one mapping of the points on the sphere to points on the plane- creating a square grid, and a honeycomb of regular hexagons. Stereographic projection is often used in representing the geography of the globe of our planet on to a flat map. Mathematical art by Henry Segerman. 


Catenary Curve

Get scale models here: 
From Amazon: BUY NOW Scale Gateway Arch 

Learn about the sunshield of the James Webb Space Telescope here:
SmarterEveryDay: Episode 270: JWST Sunshield

Click this link for more catenary curve fun!

Catenary Curve: What do the James Webb Space Telescope and the Gateway Arch have in common? The 5 sunshield panels of the JWST are engineered to be pulled into shape by the geometry of the catenary curve (see the latest video from SmarterEveryDay for amazing details on this). The Gateway arch is perhaps the most iconic use of this engineering principle and it matches the shape of a chain hanging from two ends, a curve known as the catenary or hyperbolic cosine- demonstrated here with a scale model souvenir (1”=100’). When an arch is built in the shape of this special mathematical curve the compression forces between each segment are always parallel to the curve- the arch is stable with no tendency to buckle. Famously used in design from the buttresses of Notre Dame to the Gateway Arch, and now in orbit about the Earth-Sun L2 point! 

Mirror Anamorphosis

This image by István Orosz is available as a poster and as a puzzle: 
From Amazon: BUY NOW Mysterious Island Puzzle 
From MathArtFun.com: BUY NOW Mysterious Island Poster 

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For those who want to see the math behind this art, here is an initial paper on the topic published in 2000 in the American Journal of Physics: Anamorphic Images by Hunt et al. 
Many books are available (with mirror cylinders) from Amazon: Anamorphic Art in Books 

Mirror Anamorphosis: this famous print by artist István Orosz has a hidden anamorphic image revealed by placing a mirrored cylinder over the depiction of the moon in the image. The work visualizes a scene from the book “The Mysterious Island” by the science-fiction author Jules Verne- whose portrait emerges in the reflection on the cylinder. The math describing this mapping is quite complex and was given in detail in a physics journal in 2000, but before that Martin Gardner described the math in 1975. Repost for this week’s theme as I head to G4G! 

Tessellating Geckos

Laser cut geckos are available here: 
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Tessellating Geckos: MC Escher inspired lizard cutouts interlock precisely to tile a surface with no overlaps or gaps. Laser cut from maple, walnut, and cherry wood by maker/artist Craig Caesar and inspired by MC Escher’s “Study of Regular Division of a Plane with Reptiles” 1939. G4G week: Martin Gardner wrote about the art and math of Escher in 1961- which helped create the popularity that his work has experienced ever since.


Dudeney's Dissection

This high-end (rather expensive) amazing version available here: 
From Grand Illusions: BUY NOW Dudeney's Dissection 

Click here for other intresting and afordable versions of this puzzle

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See both Wikipedia and Wolfram MathWorld for more details on the history and math of this geometrical oddity. 

Dudeney's Dissection: an equilateral triangle can be cut (dissected) into four pieces that will then assemble into a square. Interestingly the four parts are all different in shape (the green and yellow pieces are similar but not the same). This hinged model is comprised of precision machined and anodized aluminum, and can be folded back and forth between the two simplest regular polygons. It is not known if a similar three piece dissection is possible. Also called the haberdasher's problem and described in 1907 by Henry Dudeney it is the only 4 piece solution known.

Ambiguous Object 

These type of objects were invented by mathematician Kokichi Sugihara, and you can buy his other books here: 
From Amazon: BUY NOW Ambiguous Objects by Kokichi Sugihara 

Also available from Amazon (Japan): BUY NOW set of four ambiguous objects with booklet 

Similar objects available here- from Etsy: BUY NOW Ambiguous Objects

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Another illusion design by Kokichi Sugihara of Meiji University in Japan, the inventor of this illusion and art form. A mathematically calculated combination of perspective and the physics of reflection produce this striking illusion that works in many configurations.

120 Sided Fair Dice

Get one here! Many colors to choose from. 
From Amazon: BUY NOW 120 sided dice 

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he d120: mathematically this die has the maximum possible number of sides with equal area (discovered so far). Two mathematicians, Robert Fathauer and Henry Segerman, realized that the oddly named uniform convex polyhedron (disdyakis triacontahedron) had the needed geometry to make a 120 sided fair die. Like the familiar 6 sided die, the d120 has the following properties: every side must have equal area and the numbers on parallel sides (top and bottom) must sum to the same number. The inventors admit that they do not have any suggested use for these dice- they made them purely because mathematically it was possible to do so! 


Hexa Sphericon

Sphericon and Hexa-sphericon: order your set today! 
From the Matter Collection: BUY NOW The Sphericon (Hex and Regular) 

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Hexa-Sphericon: Sphericons are unique solids that roll in such a way that every point on their surface comes in contact with the plane. Solids from the sphericon family all have one side and two edges. Each sphericon is based on a regular polygon, with the basic sphericon derived from a square, and here- a more interesting case with more complex rolling motion- from a hexagon. 

Hexa Sphericon

Select metal versions, or less expensive 3D printed versions avaiable here:
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Hexa-Sphericon: Sphericons are unique solids that roll in such a way that every point on their surface comes in contact with the plane- following the path shown here with white paper. Solids from the sphericon family all have one side and two edges. Each sphericon is based on a regular polygon, with the basic sphericon derived from a square, and here- a more interesting case with more complex rolling motion- from a hexagon. 

Novascope Kaleidoscope

The Novascope can be ordered from the artist here: 
From novascopes.com: Order here Novascope by David Sugich 

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Novascope: tapered mirror kaleidoscope by David Sugich uses three mirrors to create an image of geodesic spheres. Three mirrors in an equilateral triangle configuration will produce a plane of tiled triangles, but if they are tapered the repeated reflections curve to infinity creating the spherical geometry. In this design there are thin gaps etched into one mirror which allows in colored light from a flashlight (on the white side of the pyramid shaped scope) to produce the hexagon lattice. Shining a light through the view portal reveals where the colored lines come from as a flashlight moves from top to bottom and back. Invented by the famous Scottish physicist Sir David Brewster (1781-1868), the kaleidoscope is an ultimate physics toy and entire field of artistic endeavor.


Pentominoes

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From Etsy: BUY NOW Hardwood Pentominoes 

Many versions available here: 
From Amazon: BUY NOW Pentominoes 

The book by mathematician Solomon Golomb that started the polyonomo recreational math craze:
From Amazon: BUY NOW: Polyominoes 

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Pentominoes: The 12 possible arrangements of five identical squares joined edge to edge. Since 5x12=60, the pentominoes can tile a 6 x 10 rectangle with no gaps (2339 ways to do this- yet even finding one solution is a challenge). I love this beautiful set from artist/woodworker Ron Moore where each pentomino is made from a different kind of hard wood. 

Pythagorean Puzzle

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From Creative Crafthouse: BUY NOW Pythagorean Puzzle

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Pythagorean Puzzle: a proof, in physical form, of one of the most famous equations concerning the sides of any right triangle. The area of a square with side c of the hypotenuse is indeed equal to the sum of the areas of the squares of side a and b. This kit also allows at least two other ways to prove this theorem named after the famous Greek mathematician from 500 BC. One of the most used formulas when calculating vectors in physics classes ?

The Klein Bottle

The best Klein Bottles are made by Cliff Stoll, astronomer, mathematician and artist. Every one-sided, zero volume bottle is packaged and shipped by Cliff himself. Get one today! 
From ACME Klein Bottles: Buy NOW Klein Bottles by Cliff Stoll 

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Wikipedia has great details on the Klien Bottle, and the amazing Cliff Stoll

The Klein Bottle: 3D representation of a four dimensional mathematical object with one side, no edges, and zero volume. Kind of like a Möbius strip with no edges.* Math meets glass art! Many thanks to Cliff Stoll for this kind gift and a great visit including a wonderful tour of his collection of mathematical oddities. *only achievable in 4D.