### Arrow on Mobius Strip

Get this fun demostration printed here:

From Etsy: **BUY NOW: Mobius Strip with Arrow**

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.

### Ambiguous Suits & Hearts

Get this ambigous object illusion here:

From SEO Magic: **BUY NOW: Heart Metamorphosis by Kokichi Sugihara**

See other: ambigous object illusions in my collection

Ambiguous Suits & Hearts: I first discovered and posted the work of mathematician Kokichi Sugihara’s amazing ambiguous object illusions in 2017. Here is one more really nice example of his original work that was sent to me by my good friend Tim Rowett.

### Levitating Cube Tensegrity

Get the 3D print files here: Levitating Cubes

Similar fun tensegrity prints often available on Etsy

Levitating Cube Tensegrity: connected only by four segments of fishing line! An amazing 3D print creation by @heliox_lab where the beautifully symmetric configuration is stable in all orientations. Another fun take on the tensegrity concept, invented by Kenneth Snelson and made famous by the architect Buckminster Fuller in 1949.

### Dudeney's Dissection 3D Print

Get this set here!

From Etsy: **BUY NOW: Dudeney's Dissection 3D Print **

Dudeney's Dissection: an equilateral triangle canbe cut (dissected) into four pieces that will then assemble into a square. This 3D printed version comes as a puzzle- fit the pieces in each of two containers- a square and a triangle, which also makes it clear the two supplied shapes are of equal area. Fun fact: It is not known if a similar three piece dissection is possible. Also called Haberdasher's problem and described in 1907 by Henry Dudeney it is the only 4 piece solution known.

### Symmetry Sculptures

These amazing scluptures from Bathesheba Grossman available here:

From Etsy: **BUY NOW: Math Sculptures**

Symmetry Sculptures: the Quintrino with 12 identical interlocking starfish, and the Metatrino with 6 identical connected four armed stars- based on the symmetries of the dodecahedron and the cube. Two examples of the amazing work of artist Bathsheba Grossman, a pioneer in creating such intricate mathematical forms.

### Aperiodic Monotile

Learn more about this recent math dicovery here: An aperiodic monotile (arXiv)

Althogh the math says they can tile the plane, getting them to do so is more challenging than one might think!

Get a set of laser cut hat tiles here:

From Etsy: **BUY NOW: "the Hat" monotile set**

See other aperoidic tilings in my collection.

Aperiodic Monotile: this newly discovered 13 sided shape, named “the hat”, will tessellate a plane to infinity, similar to how squares or hexagons can tile out with no gaps. However the hat tiles the plane aperiodically- if one tries to shift a part of a hat tiling, the shifted part will not align or match up with any other part of the same tiling- all the way out to infinity! The fact that aperiodic tessellations exist at all is pretty amazing, and Sir Roger Penrose (Nobel prize in physics 2020) is also famous for discovering a pair of regularly shaped polygons that tile in this aperiodic way. However it was not clear until a few weeks ago if a single shaped tile could tessellate aperiodically when the hat was described in a paper by Smith, Myers, Kaplan, and Goodman-Strauss uploaded to arXiv March 20.

### Penrose Aperiodic Rhombs

A 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.

Wikipedia has the details on Penrose Tilings and their inventor Sir Roger Penrose (Recently won Nobel Prize!)