Math Toys

The Random Walker

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

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.

The Holoscope: Icosahedron

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

Look through other holoscopes in my collection here: Holoscope Kaleidoscopes

The Holoscope Icosahedron: the intricate beauty of multiple internal reflections from 20 triangular mirrors in the shape of this famous platonic solid. The interior is viewed from one corner and illuminated by light entering from glass spheres placed at all of the other 11 vertices. A type of kaleidoscope based on mirrored polyhedra by artist Gary Allison, (swipe to see the dodecahedron and cube) and future posts will include tetrahedron and octahedron holoscope forms as I complete my Platonic solids set. Each holoscope has stained glass on the exterior and front surface mirrors on the inside which create the amazing and seemingly impossible spaces within.


Ambiguous Object Illusion Set

This wonderful and afffordabe set includes four illusion objects and a mirror:

From curiositybox.com: BUY NOW: Inq's Ambiguous Illusion Kit

Ambiguous Object Illusion Set: finally a set of these fantastic physical illusion objects available in the US for purchase. This kit comes with four objects (three shown here) invented by mathematician Kokichi Sugihara of Meiji University in Japan. Polygons appear as circles in a mirror and vice versa, and the famous “stubborn arrow” that will only point to the right (or, in a mirror, to the left). I like how the base is also an ambiguous pentagon/circle, which like all these objects, is a result of a clever combination of reflection, perspective, and viewing angle. Thanks to the Vsauce team for producing this kit! 

Hexa Sphericon

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

3D printed as well as handmade sphericons and similar shapes avaiable here:

From Etsy: BUY NOW: Sphericons 

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. 

Penrose Aperiodic Rhombs

This set avaiable here:
From : Search NOW Penrose P3 Tiling
Wikipedia has the details on Penrose Tilings and their inventor Sir Roger Penrose (Recently won Nobel Prize!) 
Another nice version of Penrose Tilings is available here: 
From Etsy: BUY NOW Penrose Tiles 

Penrose Aperiodic Rhombs: a famous aperiodic tiling with just two shapes- a pair of rhombuses with equal sides, but with the ratio of their areas made to equal the golden ratio. Note that although the starting pattern of 10 tiles is symmetrical, adding any further tiles breaks the symmetry, as highlighted by the path of the double curves. Sir Roger Penrose- who just won the Nobel prize in physics for his contributions to General Relativity- also discovered tessellations (tilings) that are aperiodic even though the two tile types are regularly shaped polygons. If one tries to shift a part of a Penrose tiling, the shifted part will not align or match up with any other part of the same tiling- all the way out to infinity! In this construction, single and double line patterns must align such that the tiles can only connect in specific ways to ensure the non-repetitive nature of the Penrose tiling structure. Shown here is one way these two tile types will fill the plane.


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 

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.


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). 

Scutoids

This set of 3D printed scutoids available here:

From Etsy: BUY NOW: Scutoid container set

Scutoids: a recently recognized form of geometric solids discovered in 2018. Scutoids have a different polygon on each end, and when packed together maximize stability and minimize energy when forming a boundary layer- such as the membranes around organs in living tissue. These 5-6 scutoids have pentagons on one end and hexagons on the other. Discovered by a collaboration of biologists and mathematicians in 2018, this set is sold by Recep Mutlu of 3DPrintBase.

Hexacon and Sphericon Rollers

Get these and other amazing developable rollers here:

From Etsy: BUY NOW: Hexacon and Sphericon Rollers

Hexacon Roller: beautiful 3D printed versions of a recent mathematical discovery of new developable rollers (objects that roll where every point on the roller’s surface comes into contact with the plane upon which it rolls). Similar to the sphericon (based on a square) the hexacon rolls in a straight line with a peculiar wobble motion but has a hexagonal cross section (swipe to see video loop of each in motion). The hexacon (2019) and sphericon (1980) are two of a family of such rollers called polycons discovered by David Hirsch, and described in a paper by Hirsch and Seaton published in January of this year. 


Sphere Sticks Geometric Puzzle

Get this affordable and amazing puzzle here:

From Etsy: BUY NOW: Sphere Sticks

Sphere Sticks Puzzle: 30 identical wood pieces, each with two notches as shown, can create 12 interlocking pentagons in a perfect symmetry- look carefully and you can see that each rod is in an identical configuration with the 4 others that connect with it. Precision cut notches on the rods allow them to interlock with elastic tension such that vector sum of the 4 forces sum to zero in this tensegrity type equilibrium. The dodecahedron, with its 30 edges and 12 sides, is the basis of this puzzle sculpture.