Math Toys

Square Dissection Puzzles

Get these and other well made disection puzzles here:

From Etsy: BUY NOW: Square Dissection Puzzles

These puzzles are expertly laser cut and sold by GamesEfce. I spray-painted the pieces of mine to better show the shapes and relationships for the video. 

Square Dissection Puzzles: a square can be cut (dissected) into polygons and then reassembled into other regular polygons. Shown here: an equilateral triangle, a pentagon, and a hexagon. These are the record holders for smallest number of pieces needed: triangle (4 pieces by Henry Dudeney 1902), hexagon (5 pieces Paul Busschop 1870s) and pentagon (6 pieces Robert Brodie 1891). Fun fact- It is not known if any of these records are the smallest possible, no mathematical proofs yet exist on this question.

Hyperbolic Holes

This inexpensive kit available here:

From Amazon: BUY NOW: Hyperbolic Holes Kit

Hyperbolic Holes: a straight rod, in this case a pencil, glides through a symmetrical pair of curved holes. The design is based on the hyperboloid, the 3D ruled surface traced by an offset rotating diagonal line. This device is sold as an inexpensive kit to assemble yourself, and includes a motor with geared drive and pre-cut pieces. The pencil is my addition- sharpened to just the right size to clear the curved openings.

Dodecagon to Square Puzzle

Get these and other well made disection puzzles here:

From Etsy: BUY NOW: Square Dissection Puzzles

These puzzles are expertly laser cut and sold by GamesEfce. I spray-painted the pieces of mine to better show the shapes and relationships for the video. 

Square to Dodecagon Dissection Puzzle: a square can be cut (dissected) into polygons and then reassembled into other regular polygons. Amazingly the 12-sided regular polygon can be cut into these six specific pieces that can then fit a square. Swipe to see the minimum cuts dissections for a hexagon, pentagon, and triangle.


Polarized Light Cell Kaleidoscope

Click here for affordable, precision made scopes with angled mirrors: Kaleidoscope Symmetries Explored

Similar Kaleidoscopes available here:

From Amazon: BUY NOW Fluid Flow Kaleidoscope 
From eBay: Fluid Flow Kaleidoscope

See more kaleidoscopes in my collection: Kaleidoscopes

Polarized Light Cell Kaleidoscope: the amazing colors you see from the object cell of this 3 mirror kaleidoscope are generated by manipulating the polarization of light- the object in the clear wand is a strip of clear cellophane in oil which has the property of optical rotation. These property allows for an incredible range of color formation when placed between two linear polarization filters (swipe for demonstration). A creation of kaleidoscope artist Ron Kuhns.

Uphill Roller

This set available here:

From Amazon: BUY NOW: Uphill Roller Double Cone

Uphill Roller: a double cone (like two funnels connect by their tops) will roll up a set of inclined rails. Although the bi-cone rolls toward the higher end, its center of mass descends due to the geometry of the rails. This curious construction was first published in 1694 by the noted surveyor William Leybourn to promote “recreation of diverse kinds” towards the “sublime sciences”. Physics fun from three centuries ago!

3D Pentominoes

The set I used for this video is called Pocket Katamino and is available here
From Amazon: BUY NOW Pentominoes 

3D Pentominoes: the 12 possible arrangements of five identical squares, joined edge to edge, form the set of all pentominoes. Since 12x5=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). This set of colorful pentominoes is made so that the height of each piece is the same as the width of the constituent squares, such that 3D constructions can be made. Since 3x4x5=60 one can build a box with these dimensions (amazingly, 3940 ways to do this- but again, finding one is still a fun challenge). 

Dymaxion Map

Get one there:

From Art of Play: BUY NOW: Dymaxion Folding Globe

Dymaxion Map: today some math fun with this unique mapping of the Earth where the globe is projected onto an icosahedron and then unfolded onto two dimensions. Invented by the famous architect R. Buckminster Fuller, the Dymaxion projection map is designed such that it does not have a “right way up” and showing the continents as “one island Earth”. This mapping also produces less distortion of relative areas and shapes- note here that Greenland looks, correctly, much smaller than Africa- unlike what is seen on many world maps where they look the same size. This version, designed by Brendan Ravenhill, uses flat magnets to allow a very satisfying transformation between the flat 2D net and the 3D icosahedron “globe”.

Soma and Rhoma Puzzles

The Rhoma Cube is a vintage item and can sometimes be found on eBay:

From eBay: Search NOW: Rhoma and Soma Puzzles

Soma Cubes are available in a wide viariety: 

From Amazon: BUY NOW: Soma Cube

From Etsy: BUY NOW: Soma Cube

SOMA and RHOMA Puzzles: Replace every cube in the Soma puzzle with a rhombohedron to create the Rhoma puzzle- where the seven slanted pieces now make a larger rhombohedron. Here I’ve painted the corresponding Soma and Rhoma pieces to match on these sister puzzles that share identical edge lengths. While there are 240 ways to make the larger cube from the 7 Soma pieces, there is only one solution for the 7 Rhoma pieces. The original Soma Cube is a math toy has an interesting connection to physics- invented by Danish scientist and artist Piet Hein who claimed that this puzzle idea came to him as he was listening to a lecture on Quantum Mechanics by Werner Heisenberg in 1933. The seven pieces are all the ways 3 or 4 cubes can be joined, such that each piece has at least one inside corner.

Arrow on Mobius Strip

Get the 3D print stl files here:

From printables: download now: Mobius Strip with Arrow

Arrow on Möbius Strip: on the geometry of a Möbius strip a right pointing arrow points left after one trip around, a second trip restores the original orientation. This mathematical property is called non-orientability, and is also true of Klein bottles which I’ve posted about. I love how this 3D printed model, designed and produced by Wes Pegden, allows one to physically manipulate and intuit this somewhat obscure mathematical property. 

Lissajous Roller 

Available from Pyrigan & Co.

From Etsy: BUY NOW: Lissajous Roller Illusion

Lissajous Roller: when viewing this 3D printed object from the side one sees a projection of a 3:2 Lissajous curve, but the object is actually cylindrical in frame and can roll towards or away from the viewer. When in motion a “dual axis illusion” is produced where the object appears to be rotating about a vertical axis. Invented by Bill Gosper and produced by Pyrigan & Co. 

Stomachion Puzzle

Get this 3-color laser cut acrylic version here:
From Kadon Enterprises: BUY NOW: Stomachion Puzzle

Also a very nice multicolor acyrlic version here:
From Etsy: BUY NOW: Stomachion Puzzle

Learn about the 1998 discovery of the lost writings of Archimedes (and the technology used to recover them) in this TED talk

Ancient Stomachion Puzzle: the oldest known puzzle, discovered in the writings of the great Greek physicist and mathematician Archimedes from some 2200 years ago. The puzzle is a dissection of a square into 14 polygons, where the areas of each piece are integer multiples of each other (a curious way to slice it up). In 2003 Bill Cutler showed that there are 536 district ways to configure these pieces to make the square (five are shown here), ignoring simple rotations and reflections. Swipe to see the most famous solution, attributed to Archimedes himself, that was found in an ancient manuscript discovered only in 1998- before this date historians knew the name of the puzzle, but no one knew what it looked like. Kate Jones, the maker of this particularly aesthetic version, found that when using only three colors for the polygons, there are only 6 solutions where no two pieces of the same color touch (four solutions shown here).

3D Wordflip Sculpture

Create your own here:

From BUY NOW: 3D Wordflip

3D WordFlip Sculpture: from one point of view my logo appears, rotate 90 degrees and the sculpture reveals my tagline phrase “real magic”. Alternatively the messages can be cast as shadows (swipe to view). Amazingly you can create and print any two phrases of your choosing from the webpage of artist and designer Joris Sparenberg.

Hexa Sphericon

3D printed as well as handmade sphericons and similar shapes avaiable here:
From Etsy: BUY NOW: Sphericons 

Sphericon and Hexa-sphericon: beautiful works of art in metal- available here!
From the Matter Collection: BUY NOW The Sphericon (Hex and Regular) 

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. 

Cone of Apollonius

Similar models available here:

From Etsy: BUY NOW: Cone of Apollonius

Cone of Apollonius: Slicing a cone with a plane will produce the famous curves known as the conic sections, as demonstrated with this beautiful vintage wood model by Nasco. Slicing at a right angle to the cone’s axis of symmetry produces a circle, and tilting the intersecting plane a bit produces an ellipse. When the plane is tilted parallel to the side of the cone the curve produced is a parabola, and tilting even further creates a hyperbola. The discovery of the mathematics demonstrated here are attributed to Apollonius of Perga from about 250 BC- thousands of years later Kepler, Newton, and others showed these conic sections to be intricately connected to many branches of physics such as planetary orbits and the optics of telescopes.

Frabjous Geometric Sculpture Puzzle

Get one here- five colors to choose from:

From MoMath: BUY NOW: Frabjous Sculpture Puzzle

Frabjous Geometric Sculpture Puzzle: 30 identical laser cut acrylic pieces interlock into 12 interconnected five point stars (each with a spiral vortex center) in this puzzle based of the Frabjous sculpture by artist and professor of mathematics George Hart (Prof. Hart is now on Instagram, follow him at @george.hart.sculptor to see more of his amazing work.) Note that if one connects the tips of the stars one gets the outline of a dodecahedron, with its 30 edges and 12 sides, and if one considers the face planes of the linked pentagrams the underlying shape is a polyhedron called the “great rhombic triacontahedron”. A year or so ago I got to visit the National Museum of Mathematics in NYC where I bought this puzzle in the @momath1 museum shop. This puzzle was great fun to assemble- buy one to support this inspirational museum, and make a great sculpture for your bookshelf! 

Rhombic Blocks Mathematical Puzzle

This beautifully made puzzle available here:: 
From Etsy: BUY NOW Rhombic Blocks 

Rhombic Blocks Mathematical Puzzle: There are 9 possible ways three rhombuses can be joined together along a common edge, and similar to pentominoes, these 9 tri-rhombs can tile a polyhedron, in this case a hexagon. There are 14 solutions to this puzzle, and one where no same-colored pieces touch. A beautiful math discovery by puzzle master Stuart Coffin.