Bakery – Mathematical Connections

The story

Arnold the Cat, aided by his able mouse assistant, bakes tessellating cookies, knotted pretzels, and other delectable treats in Mandelbrot’s Bakery.

Look closely; what do you see?

Can you find repeating patterns in…

• The mice on the bakery wall?
• The tiles on the bakery floor?
• The cookies on Arnold’s baking sheet?
• The cookie designs on the mouse’s list?

Focus on… wallpaper groups

The symmetric patterns on the blue mouse-themed wallpaper in the bakery illustrate what mathematicians call wallpaper groups. There are 17 wallpaper groups in all, classified by different types of two-dimensional symmetries. The bakery tapestry illustrates the nine groups that feature only reflections, glide reflections, and 180º rotations.

The remaining eight wallpaper groups, featuring rotations by 60º, 90º, and/or 120º, are illustrated with decorative mice elsewhere—in the floor mats in front of the Curio Shop and on the Terrace, and on the quilted wall hanging inside the Curio Shop.

Wallpaper groups describe patterns that repeat in at least two directions, while frieze groups describe patterns that repeat in only one direction.  The black and white mice on the chalkboards around the installation show the 17 two-color frieze groups.

Several decoration motifs of the Bakery are related to regular pentagons (decoration of front window and door; decoration of left side window) or 5-fold symmetry (oven door, wheel of outside cart).

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

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).

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). The unclosed circular arcs in the design of the wheel hub show a tiling of the hyperbolic disk by regular (hyperbolic) pentagons.

The part of the design of the wheel hub consisting of complete circles is an example of the fractal structure called Appolonian gasket.

Another fractal structure is shown by the heptagonal pyramids on the 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 is an example of TSP Art. 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). 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 systematically laser-cutting some segments of the original TSP path; every third segment, in this case.)

The transformation ceramic-mouse → cup → ceramic tortoise on the rim of the Bakery display cart evokes an assertion of which mathematicians are fond, that “a coffee mug is like a donut” – 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 is a painterly version of a drawing by Bill Thurston in his Geometry and topology of three-manifolds (never printed as a book version but available digitally). 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 is the result of repeated circle inversions, inspired by Indra’s Pearls.

A similar construction, with many more inversions, and closer to the original Indra’s Pearls strategy, was featured on a recent AMS calendar.

Both the decoration on the back wall of the Bakery and the doorknob feature the famous Mandelbrot set, which is also prominent on the shop sign. It is not a coincidence that mandelbrot cookies are also a specialty of the Bakery …

The intricate pentagonal window above the door shows a beautiful periodic orbit for pentagonal billiards; many more such orbits can be found in this document.

The equation on the rims of the bowls in the Bakery is the (time-dependent) Schrödinger equation, a very famous Partial Differential Equation, important in quantum mechanics. Schrödinger’s name is associated with many aspects of quantum mechanics; in particular, Schrödinger’s cat plays a role in a thought experiment designed by Schrödinger to showcase the nature of “superposition states”.

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. Arnold’s name is also linked to the so-called “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.