Math Toys

Kinetic Traced Hyperboloid

This hard to find sculpture curretnly available here:

From Amazon: Hyperbolic Kinetic Sculpture

Kinetic Traced Hyperboloid: a straight rod glides through a symmetric pair of curved holes in this kinetic sculpture based on the hyperboloid, the 3D ruled surface traced by an offset revolved straight line. This version is made of anodized aluminum and rotates via gearing and a motor powered by two AA batteries in the base. 

Satisfying Hexagons

Get this 3D print here (your choice of colors):

From Etsy: BUY NOW: Satifying Hexagons

Satisfying Hexagons: this delightful kinetic art manipulation toy features 19 nested hexagons within a hexagonal frame. Embedded magnets allows one to move the central hexagon from behind creating interesting visual effects. A 3D print created by @i.am.the.lazy.engineer- indeed oddly satisfying!

Dudeney's Dissection

Get this version here: 
From Grand Illusions Ltd: Dudeney's Dissection 

A nice wood version is available here: 
From Etsy: BUY NOW Dudeney's Dissection 

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.


Oloids: Solid and Anit-oloid

Order your Anti-Oliod today: available in three types of metal:

From The Matter Collection: ORDER NOW: Anti-Oloids in Brass, Copper, and Steel 

Oloids: “solid hull” and “ruled surface” types made from brass and copper- oloids are unique solids that roll in such a way that every point on their surface comes in contact with the plane. The basis of the oliod’s geometry is that of two connected circles, one perpendicular to the other such that the rim of each circle goes through the center of the other. The shapes you see here are the results of connecting the rims of these circles together with a family of straight lines, one method leads to the solid convex hull form, and another way leads to the ruled oloid (anti-oloid). 

Sphere and other Orbiforms

These volumes of constant width available for order now: choose from brass, copper, or stainless steel

From AltDynamic: BUY NOW: Sphere and Orbiforms

Sphere and other Orbiforms: pi day special post- volumes of constant width made from solid brass. These shapes have constant diameter no matter their orientation and will roll like spheres between two planes- note how the acrylic plate stays parallel to the table as the sphere and other orbiforms roll underneath. The first orbiform is based on the Reuleaux triangle and the second on a Reuleaux pentagon. Fun pi fact- the perimeter of any shape of constant width is alway equal to the diameter(width) multiplied by pi: P=πd.

Squaring a Circle Sculpture

This sculpture available as a 3D print:

From ShapeWays: BUY NOW: Square Circle Illusion

See other amazing geometric illusions here: Ambiguous Objects

Squaring a Circle: from one particular point of view this wireframe sculpture looks like a circle, from another it’s a square! From other viewing angles one can see that the underlying curve is comprised of four identical segments of a parabola. A wonderful example of how a single perspective can be misleading! A math sculpture available as a 3D print by Matt Enlow. 


Coins of Constant Width

These triangle coins are often available on eBay.

Fron eBay: Search NOW: Bermuda Triangle Coin

The silver proof versions can be expansive, but sometimes the circulated coins (like those in the video) are available at a lower price.

Click to see more: shapes of constant width

Coins of Constant Width: the only coins produced in the shape of the Reuleaux triangle, issued in 1997 and 1998 by Bermuda. The special convex shape of the Reuleaux triangle will roll, because like a circle they have the same diameter from one side to the other, no matter their orientation. To demonstrate this property note here how two straightedge rulers remain parallel as the coins rotate between them, just as one would expect circles to behave! Bermuda triangle (ha!) coins were only produced for two years and featured Elizabeth II on the front and shipwrecks on the back- this one depicts the wreck of the Sea Venture.

Pentominoes

Get this set here: 
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 

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. 

Wooden Trapped Sphere in Cube

Get this curiosity here:

From Art of Play: BUY NOW: The O-Cube

Trapped Sphere in Cube: a surprising aspect of the geometry of spheres and cubes- carved from a single block of wood is a sphere trapped within a cube frame. A classic folk woodworking novelty reconceived here with precision machining to create this seemingly impossible object. 


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 

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! 

Tessellation Origami

See more of Ekaterina's amazing work on her website gallery: Kusudama me! 

Contact her to buy her artwork, or you can buy her books and learn how to fold amazing geometries!

From Amazon: BUY NOW: Ekaterina Lukasheva: Papercraft and Origami

Tessellation Origami: nested spirals and triangles created from one flat sheet of paper! This beautiful work by Ekaterina Lukasheva also demonstrates how folded paper can obtain very different physical properties than that of the original flat paper. When stretched out this paper sculpture prefers to snap back into spirals and triangles, and although most materials bulge out when compressed along one direction, here the design compresses evenly along all three axis of the hexagonal symmetry.

Steinmetz Bicylinder

Three choices of metal- order one today! 

From KickStarter: ORDER NOW: Steinmetz Bicylinder

Steinmetz Bicylinder: intersect two cylinders at right angles and the remaining confined space is the bicylinder- shown here machined from stainless steel. The bicylinder casts a circular shadow along two orientations, and a square shadow perpendicular to those. In addition the curve created along where the two cylinders meet is an ellipse- as seen with the object spinning along the intersection axis. Fun fact: the area and volume of this object are known to be A=16r^2 and V=16r^3/3. Thanks to Zac Eichelberger of Math Meets Machine for sending me one of his creations. 

 

 


Hyperbola Clock

This amazing clock available here: 
From Maths Gear: BUY NOW 
Hyperbola Clock 

Hyperbola Clock: a straight rod glides through a curved hole in this unconventional clock based on the hyperboloid, the 3D ruled surface traced by rotating diagonal line. In this creation by Robert Darwen of Fibonacci Clocks, the rod serves as the hour hand with a smaller minute hand above the center of the base disk. (The time adjustment dial of the clock mechanism was connected to a small motor to produce the sped up motion in this video so that 1 second = 1 hour) 

The Holoscope: Cube with Spheres

Order a holoscope from the artist's gallery here: 
The artwork of Gary Allison: BUY NOW Holoscopeworld.com 

The Holoscope: a cube of mirrors with the interior viewed from one corner and illuminated by light entering from glass spheres at the other seven vertices. A type of kaleidoscope based on truncated Platonic solids by artist Gary Allison. Each holoscope has stained glass on the exterior and front surface mirrors on the inside which create the amazing and seemingly impossible spaces within. 

High Voltage Fractal in Wood

Amazing creations made here: 
From Etsy store EngravedGrain: BUY NOW High Voltage Fractal 

High Voltage Fractal in Wood: a Lichtenberg fractal created by a high voltage electrical current flow across a piece of wood. Since wood is an insulator a light coating of conducting water (for instance a solution of baking soda or salt) is first applied to the surface. Metal electrodes are then attached at each end of the wood piece and a dangerous source of high voltage is applied (such as a microwave oven transformer or neon light transformer). 


Mirror Anamorphic Lenticular Cup & Saucer

Message Luycho on Instagram about this design. Other designs can be seen here: 
From Luycho: 
Luycho | A New world on Mirrors 

Somtimes available here:
From Amazon: BUY NOW: Anamorphic Cups & Saucers

Click here for more Mirror Anamorphic 

Mirror Anamorphic Lenticular Cup & Saucer: a flamingo sits in a nest of flowers, revealed when the cylindrical mirrored cup is put in place. This beautiful design by Luycho uses both mirror anamorphic reflection and an accordion type lenticular dual image where turning the plate 180 degrees trades images- using my new photography turntable to nice effect. Art meets math and physics! 

Ultimate Solid of Constant Width

Available in three metals and two finishes.
From The Matter Collection: 
Order NOW: Ultimate Solid of Constant Width- Brass
Order NOW: Ultimate Solid of Constant Width- Steel
Order NOW: Ultimate Solid of Constant Width- Copper

Ultimate Solid of Constant Width: Reuleaux tetrahedrons with specially calculated curved edges become volumes of constant width- possibly the minimum volume that can possess this property. The shape featured here is a new discovery of a solid of constant width that has perfect tetrahedral symmetry. Made from solid steel, brass, and copper- these shapes have constant diameter no matter their orientation and will roll like spheres between two planes- note how the acrylic plate remains parallel to the desktop as the orbiforms roll in between. Currently available on Kickstarter from my friends at the Matter Collection : Reuleaux tetrahedrons with specially calculated curved edges become volumes of constant width- possibly the minimum volume that can possess this property. The shape featured here is a new discovery of a solid of constant width that has perfect tetrahedral symmetry. Made from solid steel, brass, and copper- these shapes have constant diameter no matter their orientation and will roll like spheres between two planes- note how the acrylic plate remains parallel to the desktop as the orbiforms roll in between. Currently available from my friends at the Matter Collection 

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