above: This is called Alexander's Horned Sphere. As you can see, the linking of the circles continues on forever.
This shape is a fantastic discovery to mathematicians in 1924, in particular, of the branch topology↗. It is interesting because it is: “An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected.”. Here's some basic explanation:
On a piece of paper, draw a simply connected closed curve (for example, a circle). Once you draw a circle, it divides the region into two sections: inside, and outside. A theorem says that this will always be so. (silly theorem, eh?) Furthermore, the two regions are similar in the sense that each is a rather coherent blob.
Now, if you have a sphere (aka ball) in space. This sphere also divides the 3D-space into two regions, the inner one, and the other one. Again, the inside region and outside region are similar in the sense that both are coherent blobs.
However, now consider this shape called “Alexander's Horned Sphere”. It divides the space into the inner region and outter region. However, the outer region is no longer a coherent blob, yet the inner region is still a un-pierced blob.
The above is a rough description on why it is interesting. To understand this exactly, a human animal needs to study math for about a decade. For a technical description, see: Alexander's horned sphere↗.
above: A shape similar to a stellated dodecahedron↗. This is actually a living animal. When it moves, the pointed tips flow freely like some sea creatures, making it extremely cute.
(This shape is not a stellated dodecahedron. Ask Seifert Surface or Bathsheba in-world if you want to know the detail)
above: A iterated function system↗ styled fractal. Note how the pattern nests.
above: A close up. Note how large it is.
above: Another fractal. The outer shape is a tetrahedron↗. Bisect each of its edges, forms a octahedron↗ in its center and 4 smaller tetrahedrons at each of the corners. Do this again on the tetrahedrons, will form this shape.
This technique of forming fractals is based on the Sierpinski triangle↗.
above: A sculpture patterned for the equiangular spiral as exhibited in Sunflower↗'s head.
There are numerous mathematical properties exhibited in sunflower heads.
For visual demonstrations of many properties of the equiangular spiral, and why it is called “equiangular”, see Equiangular Spiral. For wikipedia, goto: equiangular spiral↗.
In a sunflower's head, the count of left-twisting spirals and the count of right-twisting spirals, are always two consecutive terms in the Fibonacci sequence. For example, suppose there are 34 left-twisting spirals and 55 right-twisting spirals. The 10th and 11th terms of the Fibo sequence is 35 and 55.
A Fibonacci sequence is this sequence of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ..., where, there next term is the sum of the previous two. For example: 2=1+1, 8=3+5, 233=89+144. The sequence starts with the terms 0 and 1. For more detail, see Fibonacci sequence↗.
The study of plant growth's arrangement is called Phyllotaxy↗.
above: This sequence shows how to turn a torus inside-out. (A torus↗ is the name for the shape of a ring.)
Unless otherwise indicated, the objects shown in this page are created by Henry Segerman↗ (aka Seifert Surface in Second Life).
If you enjoyed this math in Second Life pages, you'll love Paul Bourke's Representing and modelling geometry in SecondLife↗.
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