Garden – Mathematical Connections

Garden

Mathematical Connections

If you are at the exhibit… (Garden)

The story

Chipmunks play a game with acorns and Babylonian numeral tiles, while squirrels chatter about sculptures in the garden. Meanwhile, bees and butterflies pollinate a colorful array of geometric flowers.

Look closely; what do you see?

Through their game with the acorns, the chipmunks have found a prime number. Can you see which number that is?

Focus on… prime and composite numbers

A prime number is an integer greater than 1 that can be written as a product only of itself and 1. For example, 5 can only be written as a product of 5 and 1, while 6 can be written both as a product of 6 and 1 and as a product of 3 and 2. Six is an example of a composite number

The chipmunks are sorting prime and composite numbers. They have found many ways to arrange 12 acorns evenly in rows, and have determined 12 to be composite. But, there seem to be no such arrangements for 13–it must be prime.

In the meadow, the squirrels take a more advanced approach to finding primes using a Sieve of Eratosthenes. They have already eliminated multiples of 2, 3, and 5 using translucent sieves. Meanwhile, the sieves for 7 and 11 are rolling in…

If you are at the exhibit… (Reef)

The story

The bay and tidal pools are alive with aquatic creatures, hyperbolic corals, and polyhedral shells.

Look closely; what do you see?

A polyhedron is a three-dimensional form having flat, straight-edged faces. In the reef–and throughout Mathemalchemy–are polyhedra made from various materials. Can you spot some disguised as…

  • barnacles on the piers?
  • pearls inside seashells?
  • stamens and pistils in flowers along the shoreline?
  • pebbles on the beach?
  • jellyfish swimming in the bay?
Focus on… polyhedra

Mathemalchemy’s 3D-printed polyhedra represent four categories of polyhedra: the 5 Platonic solids, the 13 Archimedean solids, the 92 Johnson solids, and 13 of infinitely many prisms/antiprisms. These are all convex, meaning that any line drawn between two points in the shape stays entirely within the shape. 

  • A Platonic solid is a convex polyhedron in which the faces are identical regular polygons. Regular polygons have equal sides and equal angles. A cube is an example of a Platonic solid. It has 6 square faces, 12 equivalent edges, and 8 equivalent vertices. 
  • Prisms and antiprisms have a translated regular polygon as their top and bottom, and all other faces are identical parallelograms that join the two bases. In antiprisms, the polygonal sides alternate in orientation, whereas in prisms they have the same orientation. The prisms used in Mathemalchemy have rectangular faces between the bases, while the antiprisms have triangular faces.
  • An Archimedean solid is a convex polyhedron where each face is one of at least two different regular polygons that meet in identical vertices. (Usually, the prisms and antiprisms are excluded from the Archimedean solids.)
  • A Johnson solid is a convex polyhedron made from at least two different regular polygons, but with no restrictions on which polygons meet at a vertex. (Usually, the prisms,  antiprisms, and Archimedean solids are excluded from the Johnson solids.) A square-bottomed pyramid is a Johnson solid. It has one square side and four triangular sides. The vertex at the top of the pyramid, where the four triangles meet, is different from the four vertices where the square and two triangles meet.

Another category of polyhedra, called Catalan solids, inspired the embroidery designs in the Ball Arches.

Can you see how these different categories of polyhedra are rendered differently in Mathemalchemy?

The young chipmunks engage in play, sorting primes on a grid playground using acorns to explore factors, while holding clay tablets with Babylonian cuneiform numerals. They discover which numerals are prime and which are composite through brute force with their limited tools: for each number they count out exactly that many acorns and then see whether they can arrange them in rows of a fixed length, checking whether that works out evenly or leaves a remainder.

Sieve of Eratosthenes

The squirrels are taking a more advanced approach with their Sieve of Eratosthenes.

They first realize that as the smallest integer exceeding 1, the number 2 has to be prime; since no non-trivial multiple of 2 can be prime (since it has 2 as a diviser), they “sieve” those out (by inserting a screen that veils them). The next smallest number, 3, is still unveiled when they count up to it, so it is prime as well, and now they veil all its multiples with another screen. As they go higher and higher, skipping numbers that are already veiled, they discover the primes one by one, continuing to eliminate multiples of each new prime. In the present scene, the screens for 2,3 and 5 have been inserted, and the screens for 7 and 11 are approaching. Since the squirrels are only checking up to 100, and all the numbers from 8 to 10 (the square root of 100) are already veiled, the sieve for 11 is in fact not needed: all the multiples of 11 below 100 will already have been sieved out after the earlier rounds. At that point, all the numbers below 100 that are still unveiled are prime.

Animated sieve of Eratosthenes
Gaussian integers

The not-uniformly brown pavement stones in the garden path represent the result of a similar Erathostenes-sieving for the Gaussian integers, a complex analog of regular integers, consisting of all the numbers of the form m+ni (where i2=-1).

The pavement stones show an individual tiny tile for each Gaussian integer with |m| and |n| not exceeding 7, with the tile for m=n=0 in the dead center. On one of the non-uniform brown stones, only four tiny tiles are white — these are the four roots of 1 (namely, 1, -1, i and -i); they don’t count in our hunt for Gaussian primes (just like 1 doesn’t count when one lists the natural number primes); these are the only ones for which m2+n2=1. The next possible higher value of m2+n2 is 2, when |m| and |n| both equal 1. There are four corresponding Gaussian integers, namely 1+i and the results of multiplying 1+i with i, -1 and -i (the other roots of unity), giving 1-i, -1+i and -1-i. These are thus the first Gaussian primes we find; their non-trivial multiples are all the Gaussian integers in which |m|+|n| is even and different from 2 — the second brown-and-white paver shows all those numbers in white, as well as the center zero and the four roots of unity: they have been “sieved out”, and only the remaining brown tiles could be candidate primes. (Note that this means the ordinary integer 2 has been sieved out: since 2=(1+i)(1-i), it is NOT a prime in the Gaussian integers!) Next up, as we look to higher values for m2+n2 among the still-brown tiles, are 2+i (and the results of multiplying this with roots of unity, -1+2i, -2 -i and 1-2i) as well as its complex conjugate 2-i (and 1+2i, -2+i, -1-2i). The third not-all-brown paving stone therefore retains all the white tiles from the previous paver, and also blanks all the multiples of 2+i, such as 3+4i=(2+i)2 or 5=(2+i)(2-i), which were still brown after the previous step; the next not-all-brown stone after that blanks out also the multiples of 2-i. And so on … but at the scale of our pavers, nothing will change from now on: the next prime is 3, but all its multiples for which neither |m| nor |n| exceeds 7 have already been removed. So the fifth not-all-brown paving stone is identical to the fourth: the sieve for 3 is superfluous at the scale we are seeing on the pavers, just like the one for 11 was for the big upright 1-to-100 sieve for the standard integers.

Hexagonal tiling

The Riemann cliffs form one of the boundaries of the garden; they are hexagonal vertical columns, following a hexagonal tiling for their ground plan. Hexagonal tilings occur in nature in many places; a familiar example is the arrangement of cells in a beehive, as on the back of the squirrels’ sieving frame.

This particular beehive illustrates the selfsimilar nature of hexagonal tiling by displaying four successive scales : each coarser layer is obtained by enlarging by a factor √3 a finer-scaled layer and rotating by 30 degrees; the centers of all the larger cells can then be made to coincide exactly with the centers of (one in four) of the smaller cells.

The sheet that escaped from the Cavalcade fills in a hexagonal-tiling structure with the first few rows of Pascal’s triangle — note that the odd entries mark a Sierpinski triangle!

Non-euclidean geometry

Both the Garden and the Reef have many examples of undulating surfaces that illustrate hyperbolic geometry, an example of non-euclidean geometry. There are two types of non-euclidean geometry: spherical and hyperbolic. On a sphere (a surface with constant positive curvature) triangles (figures enclosed by walking from one point A to another point B by the shortest path possible, continuing similarly from B to a third point C, and then back to A) have three angles that add up to more than 180 degrees — unlike the plane (with its euclidean geometry) where the three angles in a triangle always add up to 180 degrees. This is one of the characteristics of a surface of positive curvature; another way to detect its non-euclidean nature is to realize that the circumference of a circle with a given diameter is less than π times that diameter.

  • Coral reef and see creatures in Knotical Scene
  • Knotical Scene - reef with crochet coral, beeded starfish

On the hyperbolic plane (a surface of constant negative curvature), which also has a non-euclidean geometry, the situation is the opposite from the spherical case: the three angles of a triangle add up to less than 180 degrees, and the circumference of a circle with diameter d exceeds π times d. In both spherical and hyperbolic geometry the 5th postulate of Euclid is violated: in the spherical case, whenever one picks a shortest-path line and a point P outside that line, walking straight from P into any direction eventually leads to an intersection with the line — there is thus no parallel through P to the original line; in the hyperbolic case, there can be several, as illustrated on the large yellow hyperbolic surface in the garden, where straight lines are picked out in red, and one can see three distinct lines going through the same point, none of which intersect another line.

Another artifact illustrating hyperbolic geometry is the green-and-yellow checkered fabric draped over the bottom of the hill, behind the chipmunks. On a chessboard, squares in alternative colors meet in foursomes at the grid defining the chess pattern; at each corner point, their 4 right angles add up to the expected 360 degrees.

On the flouncing fabric here, the polygons meeting in each four-corner-point are regular pentagons; their total angle sum of (about) 514 degrees makes it impossible for the fabric to lie flat — this is again a surface with negative curvature. More information about hyperbolic geometry and negative curvature surfaces can be found here.

Surfaces with negative curvature occur a lot in nature — examples are some coral reefs and mushrooms, or, closer to home, the interior surfaces of our intestine; they are the natural solution if you want to make a lot of 2D surface area in a small 3D region.

Alexander’s horned sphere

The metal sculpture in the Garden represents Alexander’s horned sphere, an important counterexample from the early 20th century. In 2D the region enclosed by a non-self-intersecting closed curve and the region outside it (the remainder of the 2D plane) can always be made to correspond continuously to the inside and outside regions of the unit circle. Before Alexander’s construction, it was believed that the same might be true for all non-self-intersecting closed surfaces without boundaries in 3D, i.e. that if the region enclosed was simply connected (meaning that any closed curve completely within the region could be shrunk to a point in a continuous fashion, without leaving the region), it and the remaining region outside would correspond similarly to the inside and outside of the unit sphere. The Alexander horned sphere example showed this belief was mistaken: its inside region (the 3D region enclosed by this surface) can be mapped continuously to the inside of a unit sphere, but its outside region cannot be mapped continuously to the region outside the sphere.

Alexander’s Horned Sphere in the Garden
Alexander’s Horned Sphere
Johnson solids

All through the Garden and Reef scenes there are many examples of regular polyhedra — the platonic and archimedean solids are all there, as are many of the Johnson solids (the remaining Johnson solids are in other scenes). The Johnson solids are realized as solids of which only the edges are indicated; the archimedean solids have some but not all of their facets filled-in, and the 5 platonic solids have all their faces filled-in.

Origami

The Garden and Reef also contain many origami objects. Traditional origami starts with a single sheet of paper; by following very precise sequences that involve many actions of folding, unfolding, refolding, complex shapes can be realized. Several of the shapes in the garden and reef (such as the roses) are origami objects of this type. Many others are examples of modular origami, in which one starts with many sheets of paper; after folding these individual sheets into identical modules, the modules are deftly assembled to make complex highly symmetrical objects, appealing to anyone who likes mathematics or symmetry. More information about the links between mathematics and origami can be found here.

  • Andrea's Rose origami in Mathemalchemy
  • Li-Mei's Flowertowers in origami for Mathemalchemy
  • Rocks
  • origami

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