**Explore the many mathematical concepts in the bakery**

## Design in Geometry

Several decoration motifs of the Bakery are related to regular pentagons:

- decoration of front window and door
- decoration of left side window
- 5-fold symmetry (oven door, wheel of outside cart).

The oven, sandwiched between the store front of the pentagon-themed Bakery and the heptagon-themed Lighthouse, uses heptagonal (a.k.a. 7-gonal) motifs in its sidewall and roof.

## Symmetry in the Tapestry

The mouse wallpaper illustrates 9 of the 17 wallpaper groups; these are the 9 groups that feature only reflections, glide reflections and 180 degree rotations, which makes them easy to implement with knitting (the craft technique used for this wallpaper).

The remaining 8 wallpaper groups are illustrated elsewhere in Downtown (near the Curio Shop and on the Terrace).

## Tessellation

### Tiling Cookies

The pi-cookie shape tessellates the two-dimensional Euclidean plane, as do the other shapes proposed by Mo[u]se.

**A different tessellation is illustrated by the floor tiling, which is one of the pentagon tilings discovered by Marjorie Rice.**

A tiling of a different nature can be found on the wheel of the Bakery display cart (against the outside wall) (picture: Mathemalchemy-Cart Wheel.jpg in After_March_2020_Workshop/Downtown/Bakery/Outside_cart). The unclosed circular arcs in the design of the wheel hub show a tiling of the hyperbolic disk by regular (hyperbolic) pentagons (picture:pentagonal_tiling_disk.png in After_March_2020_Workshop/Downtown/Bakery/Outside_cart).

The part of the design of the wheel hub consisting of complete circles is an example of the fractal structure called Appolonian gasket (link: en.wikipedia.org/wiki/Apollonian_gasket). Another fractal structure is shown by the heptagonal pyramids on the oven roof (picture:20210508_084301.jpg in After_March_2020_Workshop/Downtown/Bakery/Oven_roof). (In both cases, only the first few iterations are shown of the construction that would build the fractal if continued ad infinitum.)

The design of the grille of the furnace of the oven (picture: already present online in shorter write-up) is an example of TSP Art. (link: www2.oberlin.edu/math/faculty/bosch/tspart-page.html) Given a large number of points in the plane, finding the shortest closed path that visits each point is called the Traveling Salesman Problem (or TSP) (link: en.wikipedia.org/wiki/Travelling_salesman_problem). This NP-hard problem has been studied extensively. The designer of the grille imposed that a celtic-knot type region had to be mostly avoided; once the shortest path was found, some of the path segments were erased to create a grille design with many connections. (Without any deletions, laser cutting along the path, which is a Jordan curve, would have led to the “inside” part falling out at the end! For structural stability, it was better to omit laser cutting many segments of the original TSP path; every third segment, in this case.)

The transformation ceramic-mouse → cup → ceramic tortoise (picture: IMG_5967.HEIC in Construction/July_13) on the rim of the Bakery display cart evokes an assertion of which mathematicians are fond, that “a coffee mug is like a donut” (link: en.wikipedia.org/wiki/File:Mug_and_Torus_morph.gif) – topologically speaking at least: the boundaries of each of these two physical objects can be viewed as a closed surface with one hole/handle, which makes them topologically equivalent. The ceramic mouse and tortoise both have the same property again (their eyes are represented by a puncture in the clay from side to side; this makes the hole), so they are both similar to a cup.

The picture on the wall of the Bakery (picture: train_tracks_in_bakery in After_March_2020_Workshop/Downtown/Bakery) is a painterly version (link: conan777.wordpress.com/2012/04/22/traintrucks/) of a drawing by Bill Thurston in his *Geometry and topology of three-manifolds *(never printed as a book version but available digitally here [link library.msri.org/books/gt3m/]. These train tracks are special geodesic paths on hyperbolic manifolds that are tangent and go off in different directions – if you want to know more, read Thurston’s book!

The design on the cast-iron oven door (picture: Mathemalchemy-Oven-Door.jpeg in After_March_2020_Workshop/Downtown/Bakery) is the result of repeated circle inversions, inspired by *Indra’s Pearls* [link: en.wikipedia.org/wiki/Indra’s_Pearls_(book)]. A similar construction, with many more inversions, and closer to the original Indra’s Pearls strategy, was featured [link: www.ams.org/publicoutreach/math-imagery/burns] on a recent AMS calendar.

Both the decoration on the back wall of the Bakery (picture:IMG_1422.jpeg in Construction_July_2021/Edmund (all through construction)/Final construction) and the doorknob (picture:Mandelbrot_set_doorknob.jpeg in After_March_2020_Workshop/Downtown/Bakery) feature the famous Mandelbrot set (link: en.wikipedia.org/wiki/Mandelbrot_set), which is also prominent on the shop sign (picture:shop_sign.jpeg in After_March_2020_Workshop/Downtown/Bakery). It is not a coincidence that mandelbrot cookies are also a specialty of the Bakery …

The intricate pentagonal window above the door (picture:shop_sign.jpeg in After_March_2020_Workshop/Downtown/Bakery) shows a beautiful periodic orbit for pentagonal billiards; many more such orbits can be found in (link: arxiv.org/pdf/1810.11310.pdf).

The equation on the rims of the bowls in the Bakery (picture: already present online in shorter write-up) is the (time-dependent) Schroedinger equation (link: en.wikipedia.org/wiki/Schrödinger_equation), a very famous Partial Differential Equation, important in quantum mechanics. Schroedinger’s name is associated with many aspects of quantum mechanics; in particular, Schroedinger’s cat plays a role in a thought experiment designed by Schroedinger to showcase the nature of “superposition states” (link: en.wikipedia.org/wiki/Schrödinger’s_cat).

Finally, the names of Baker and his Assistant, Arnold and Mo[u]se, are a wink to the famous mathematicians Arnold and Moser, who worked on Dynamical Systems; one of the celebrated results to which both their names are linked is the Kolmogorov-Arnold-Moser (or K.A.M.) theorem (link: en.wikipedia.org/wiki/Kolmogorov–Arnold–Moser_theorem). Arnold’s name is also linked to the so-called “cat map” (link: en.wikipedia.org/wiki/Arnold’s_cat_map), an example of a mapping from the square to itself that has mixing properties; this map is illustrated elsewhere in the installation, in the Cavalcade.