### Hexa Sphericon

Sphericon and Hexa-sphericon: order your set today!

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. 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.

### Vsauce Mirror Anamorphosis

Currently this mirror and image set comes to you (in the Summer 2020 box) with any subscription:

From the Vsauce team: **BUY NOW: The Curiosity Box**

The Curiosity Box is an excellent way to start your own physics toy collection- reccomended highly!

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.

Vsauce Mirror Anamorphosis: the warped printed image is restored only with a mirror in the form of cylinder- when placed carefully in the center of the image, Kevin, Michael, and Jake appear in their true form. This mirror (along with a set of images and DIY templates) came to me last week in the most recent @thecuriositybox- a fantastic way to start your own physics toy collection. The math describing this anamorphic mapping is quite complex and is nicely detailed in a physics journal article from 2000.

### 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

### 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.

### 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.

### Orbiforms

Latest orbiforms available here:

From Kickstarter: **Order NOW **

Orbiforms in Steel, Brass, or Copper

Orbiforms: volumes of constant width 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 stays parallel to the table as the orbiforms roll underneath. The first set shown are based on the Reuleaux triangle and the second set are based on a Reuleaux pentagon. Currently available on Kickstarter from my friends at @altdynamic** **

### Reuleaux Rotor

This toy is availble from Amazon Japan and will ship to the US:

From Amazon.jp: **BUY NOW: Reuleaux Rotor Wodden Toy**

Reuleaux Rotor: this famous curve of constant width, the Reuleaux triangle, can rotate such that at all times it remains in contact with all four sides of a square. As demonstrated by this wooden toy from Japan, the rotor covers approximately 98.77% of the area of the square, missing only the sharp corners. The curves in the corners are in the shape of an elliptical arc. Fun fact: a Reuleaux triangle has a perimeter equal to pi times its width- just like a circle!

### Pocket Scintillator Kinetic Art

Logan sometimes has items for sale here:

From Etsy: **BUY NOW: PocketScintillators**

Pocket Scintillator Card: three sheets of seemingly random arrays of translucent colored pixels produce words and images when stacked- shift the stack of sheets and a second images appears! Innovative kinetic optical art by inventor, artist, software developer Logan Kerby @thanksplease who kindly sent me these cards encoded with @physicsfun themes.

### Novascope Kaleidoscope

The Novascope can be ordered from the artist here:

From novascopes.com: **Order here Novascope by David Sugich **

Novascopes can sometimes be found on eBay

From eBay:** Search NOW Novascope Kaleidoscopes **

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.

### 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!

### Cone of Apollonius

Similar models available here:

From Amazon: **BUY NOW: Cone of Apollonius**

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.

### 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.

### Shadow Stereographic Projection

These mathematical art objects are created by Henry Segerman and available here:

From Shapeways:** BUY NOW Mathematical Art **

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.

### Equilateral Triangular Kaleidoscope

This inexpansive kaleidoscope is available here:

From increadiblescience: **BUY NOW: Moire Tube Kaleidoscope**

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

See more kaleidoscopes in my collection: Kaleidoscopes

Equilateral Triangular Kaleidoscope: three mirrors arranged in a 60-60-60 degree triangle creates the appearance of a plane filled with triangles (or equivalently a honeycomb lattice)- perhaps the most common mirror configuration design, this inexpensive kaleidoscope produces an excellent example of the reflection pattern. As a bonus the exterior tube on this scope incorporates a kinetic Moirè pattern. The kaleidoscope was invented by the famous Scottish physicist Sir David Brewster (1781-1868), and has become an entire field of artistic endeavor.

### 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!

### Tapered Mirrors Kaleidoscope

This design by Koji Yamami available here:

From kaleidoscopeshop.com: **BUY NOW Space Teleiedoscope **

Click on this link for details on the physics and symmetries of two mirror kaleidoscopes.

Tapered Mirrors Kaleidoscope: the unique design of this teleidoscope uses three mirrors to create an image of a geodesic sphere. As can be seen through the semi-transparent acrylic tube, the three mirrors are tapered, with their smaller ends near the ball shaped lens. 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 sphere. In this design by Koji Yamami there are small gaps between the mirrors which allows in colored light from the iridescent tube to produce the radiant streaks of light. Invented by the famous Scottish physicist Sir David Brewster (1781-1868), the kaleidoscope is an ultimate physics toy and entire field of artistic endeavor.

### Square Kaleidocycle

This book has many versions of kaliedocycles: cut out and glue to make many interesting mathematical objects.

From Amazon: **BUY NOW: MC Escher Kaleidocyles**

Square Kaleidocycle: a ring of eight linked tetrahedra. The hinged connections allow the ring to be rotated through its center. The faces of the pyramids are decorated with the famous tessellation work of MC Escher, a pattern of interlocking lizards. Note that as the kaleidocycle is rotated the lizards at the center change through each of four colors. Made from card stock, this kaleidocycle was cut and assembled from a book by mathematicians Doris Schattschneider and Wallace Walker.