## Curio Shop

### Mathematical Connections

## If you are at the exhibit…

#### The story

Inside the Curio Shop, Harriet the Bowerbird curates a collection of visually and mathematically beautiful objects.

## Look closely; what do you see?

The compact works of art displayed throughout the Curio Shop can be described in terms of the mathematics that inspired them—yet it remains difficult to discern why certain mathematical connections and proportions generate greater aesthetic appeal than others.

Do some objects in the Curio Shop stand out to you more than others? What sets them apart for you?

## Focus on… spirals

One figure with a long history in art and architecture is the* Golden Spiral*, a kind of spiral that becomes wider, every quarter turn, by the numerical factor φ = 1 + √5/2. The Golden Spiral appears in a few places in Mathemalchemy, including inside the Curio Shop and on one of the pages in the Cavalcade. (Can you find them?)

The Curio Shop prominently features another mathematically-derived spiral: the *Harriss Spiral*. This spiral is derived in a way similar to the Golden Spiral, but instead of iteratively dividing a rectangle into a square plus a similar rectangle, it iteratively divides a rectangle into two similar rectangles plus a square.

Can you find Harriss spirals in three different places?

The Curio Shop is named Conway’s Curios, in tribute to famous mathematician John H. Conway, who died in 2020 from Covid-19.

## Harriss curve

The curve spiraling around the name of the shop, and forming the first and last letters on the sign, is the (fractal) Harriss curve; it is featured much larger on one of the side walls within the shop.

This curve is related to the plastic number, and for this reason its creator, Edmund Harriss, also calls it the plastic spiral, in contrast to the Fibonacci or golden spiral, linked to the golden ratio, depicted on the tiles and coasters displayed near the door of the shop.

## Abacus & Borromean rings

The shopkeeper is a fanciful bower bird, well known for its predilection for collecting beautiful objects. This bird, from a previously unknown species, is decorated with small Harriss spirals. Her name, Harriet Conway, is a nod not only to Conway but also to Edmund Harriss; after all, she has Harriss spirals all over her back! Her perch is a wooden stand showing several iterations of an Iterated Function System (IFS); if used at infinitum, the result of this IFS would be a fractal tree. The shopkeeper’s calculator, standing on the ground next to her perch, is an abacus. Borromean rings, next to the abacus, are an interesting concept (that dates back far in history): three rings, entangled in such a way that if any of the three is removed, the other two are no longer linked to each other.

## Origami ovoid, Archimedean solids, Klein bottle & Conway knot

A selection of fascinating mathematical delights is displayed in the left window of the shop. From left to right and top to bottom, they are the following: an origami ovoid; a beaded polyhedral structure, the facets of which form a rhombicosidodecahedron, one of the Archimedean solids; a many-layered and many-holed geometric object with dodecahedral symmetry; a (short) beaded DNA segment: a double helix where the vertical distance between the strands is a little less than half the height climbed by each strand in a complete turn (but there is much more to the geometry of DNA); a metal Klein bottle; and a realization of the Conway knot – the solution in 2018 by Lisa Piccirillo of an open problem related to this knot made a big splash in the mathematics world.

## Wallpaper group

The mat in front of Conway’s Curios illustrates one of the wallpaper groups missing from the Bakery wallpaper. Three of the 17 wallpaper symmetry groups involve rotations by 90 degrees and can thus be represented well by cross-stitch embroidery; this is one of those three – the other two are on the terrace on top of Conway’s Curios.

The remaining 5 wallpaper groups involve rotations by 60 or 120 degrees, and their hexagonal or triangular symmetry lends itself better to fabric piecing and quilting techniques than knitting or cross-stitch embroidery; they are illustrated on the wallhanging in this same shop.

## Moebius band & Alexander Horned sphere

At the top of the front window a ceramic Moebius band hangs next to an approximation of Alexander’s Horned Sphere, stopped after a finite number of iterations to make fabrication possible. Both the Moebius band and the Alexander Horned sphere are famous examples in the topology of manifolds.

## Geometry

At the bottom of the front window a small hyperbolic surface nestled on a torus sits next to a larger torus halved with one cut into two linked halves, another topologically neat object. Next to it stands a hyperboloid-like structure; if a tiny creature holding a teensy laser pointer were walking the top heptagonal rim while keeping the pointer beam aimed at its equally tiny friend who is simultaneously walking on the window sill, right near the bottom rim, always an angle of 360/7 degrees ahead of the top rim walker, then the laser beam would describe this ruled surface. It was cut from a coarse-grained brick using very narrow high-powered (and straight!) water jets.

## Hopf fibration

At the top of the narrow right-most window of the shop hangs a mathematical sculpture illustrating the Hopf fibration of the 3-sphere.

## Voronoi cells

The pavement outside the Bakery and Curio Shop, continuing to the bottom of the Lighthouse, is a tiling by Voronoi cells, defined by the points indicated by copper “nails”.

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