### Four Mirror Kaleidoscope

Inexpensive 3-mirror kaleidoscopes are available here:

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

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

See more amazing kaleidoscope designs from my collection: Kaleidoscopes

Four Mirror Kaleidoscope: a precise square arrangement of mirrors produces a four fold symmetry that “wallpapers” the view field in this unique handcrafted kaleidoscope design by Peter Roberts. Swipe to compare to a more traditional three mirror design. Invented by the famous Scottish physicist Sir David Brewster (1781-1868), the kaleidoscope is an ultimate physics toy and entire field of artistic endeavor.

### Interactive Logic Gate Display

Get this device here:

From Etsy: **BUY NOW: Interactive Logic Gate Display**

Interactive Logic Gate Display: the most basic four binary/Boolean logic operations that are the fundamental components of modern computers- presented here appropriately on a standard green printed circuit board (PCB) complete with the truth table for each device. These logic gates are the basic building blocks of any logic circuit, from multiplexers, arithmetic logic units, and computer memory to full microprocessors which may contain hundreds of millions of such gates on a microchip. An educational, elegant, and fun design by Tyler Jacobs.

### 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 recent 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

### 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.

### Tapered 4-Mirror Kaleidoscope

These prescision kaleidoscopes are crafted from front surface mirrors and other specialty materials, many with innovative desgin and construction techiques:

From Etsy: **BUY NOW: Kaleidoscopes by Marc Tickle**

Tapered 4-Mirror Kaleidoscope: not CGI- these images are produced by mirrors and light! Hidden within this colorful tube are 4 mirrors, 2 rolling glass spheres, and 1 dichroic glass reflector. These elements combine to produce the dynamic and magical images seen through the view window as the device is slightly tilted back and forth and brought in and out of a bright light source. Entitled “Dichroic Disco” this piece is but one of many amazing creations by artist Marc Tickle, taking kaleidoscope design to the next level. Invented by the famous Scottish physicist Sir David Brewster (1781-1868), the kaleidoscope is an ultimate physics toy and entire field of artistic endeavor.

### Inverting Valentines Illusion

This 3D printed illusion available here:

From Shapeways:** BUY NOW Red Hearts Illusion Object **

Similar item here:

From Etsy: **BUY NOW: Red Hearts Illusion **

Inverting Valentines Illusion: Love is a matter of perspective in this illusion that depends on viewing angle. A sweet take on the ambiguous object illusion invented by Kokichi Sugihara of Meiji University.

### Catenary Curve

Get this scale model here:

From eBay:** 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!

### 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 2020.

### Sphere and other Orbiforms

Similar items available here:

From Etsy: **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.

### Orbiforms

Orbiforms available here:

From Etsy: **BUY NOW Orbiforms**

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. Featured items from @altdynamic** **

### Curves of Constant Width

Get a set here:

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

Click here for 3D **Solids of Constant Width**

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).

### 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: this post celebrates pi day with coins in the form of Reuleaux polygons- shapes for which their perimeter/diameter = π, just like the circle! The special convex shapes of the Reuleaux triangle (Bermuda 1 dollar) and Reuleaux heptagon (UK 50 pence) 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 how two straightedge rulers remain parallel as the coins rotate between them, just as one would expect circles to behave. Bermuda triangle (ha!) coins are the only coins produced in the shape of the Reuleaux triangle, issued in 1997-98 and featured Elizabeth II on the front and shipwrecks on the back.

### Hyperboloid Spinner

Kit available here:

From Amazon: **BUY NOW **

Hyperboloid Spinner: HypnoGizmo

Hyperboloid Spinner: the HypnoGizmo toy consists of a set of slanted straight nylon lines arranged to form the outline of a hyperboliod- the quadratic surface related to the revolution of hyperbola around its axis of symmetry. As the device rotates the beads slide along in succession on one of the straight paths leading to the complex visual display. So much fun math in this toy!

### Squaring Circles

This sculpture available as a 3D print:

From ShapeWays: **BUY NOW: Square Circle Illusion**

See other amazing geometric illusions here: **Ambiguous Objects**

Squaring Circles: from one particular point of view these wireframe sculptures looks like a circles/squares, from another it’s a square/circle! From other viewing angles one can see that the underlying curves are four identical segments of a parabola conjoined. Further examples of how a single perspective can be misleading! Math sculptures available as a 3D print by Matt Enlow.

### Tension Integrity Icosohedron: Tensegrity

A nice version of this tensegrity icosohedron is sold as a toy for tiny tots:

From Amazon:** BUY NOW Tensegrity Toy **

I constructed this version by referring to the images and descriptions of tensegity on Wikipedia

Tension Integrity Icosohedron: Six brass struts float isolated from each other but held in a stable configuration by a net of 24 connecting cables. I made this sculpture using hollow brass tubes and weaving through them a single strand of fishing line, which is connected after passing through each tube exactly four times. This configuration of three sets of parallel struts forms a Jessen’s icosahedron under tension, and was invented by the famous architect Buckminster Fuller in 1949.

### The iTOP

Well engineered brass spinning toys by SiriusEnigmas:

From Etsy: **BUY NOW: The iTOP **

From Etsy: **BUY NOW: The PhiTOP**

From Etsy: **BUY NOW: The eTOP**

From Etsy: **BUY NOW: The PiTOP**

The iTOP: inverting spinning disk- the equilibrium state for a spinning thing is often different from that of the same object when stationary. When spun, the iTOP almost instantly inverts to raise its center of mass (shown in slow motion because it happens so fast). However, when the rotation speed decreases to a certain rate the system becomes unstable (shown again in slow motion) and flips again before going into a rolling/spinning motion like a coin. This top completes the set of four spin tops by astrophysicist Kenneth Brecher, all made of polished brass and themed on a mathematical constant. Swipe to see a demonstration of each: iTOP (square-root of -1), PhiTOP (golden ratio, φ), eTOP (base of the natural log), and the PiTOP (C/D of a circle, π).

### The PiTOP

Well engineered brass spinning toys by SiriusEnigmas

From Etsy: **BUY NOW: The PiTOP**

From Etsy: **BUY NOW: The PhiTOP **

The PiTOP: this beautiful brass cylinder has a radius of 1 inch, a height of 1/π inches, and displays the first 109 digits of π on its face. Spin this disk (best with sound on) and it will demonstrate some very interesting physics involving energy transfer and conservation of angular momentum. The edges of the cylinder are rounded and engineered to exhibit an optimum motion of a “spolling” coin, a motion that combines spinning and rolling (closeup shown in 240fps). As the disk’s angle of inclination decreases the speed of the rolling increases dramatically until the contact point with the mirror is moving in excess of 200mph. From the mind of physicist Ken Brecher, inventor of the PhiTOP (swipe for video). Note also that with the specified height, that the volume of this cylinder is exactly 1 cubic inch!

### High Voltage Wood Burn Fractal

Beautiful wood fractal art available here:

From Etsy: **BUY NOW: High Voltage Fractal Art**

High Voltage Wood Burn Fractal: a classic Lichtenberg fractal created by electrical discharge- in this piece artist/maker Nic Ferretti applied cobalt blue epoxy resin to fill the cavities of the burn pattern and create the look of frozen lightning. Swipe to see my attempt at this process with an old neon sign high voltage transformer using 7500 V at 20 mA. I turned the power off right before the two sides connected to get the fern-like look. The wood is slightly wetted with baking soda dissolved in tap water. WARNING high voltage can kill- do not use a transformer like this unless you are fully trained in electrical high voltage safety.