Garden – Mathematical Connections

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.

The story

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

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.

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.

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!

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.

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.

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.

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.

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.