There is a fun calculation for finding the geodesic of a sphere, but for most of us it is a lot easier to imagine slicing a grapefruit into two perfect halves to expose its moist geodesic surface.
A great circle lies on the plane that divides the sphere into two equal halves, just as the equator does. The great circle is visible when you make a “fruit stamp” from the wet face of one of the grapefruit halves onto a piece of newsprint.
Every circle in Euclidean 3-dimensional space is a great circle of exactly one sphere, and every one of these great circles is called the geodesic of its corresponding sphere. The flat, printable face forms the image of the geodesic in 2D.
Try stamping the round face (great circle) of your grapefruit into a group of many circle-stamps, and notice the resulting arrangement. It is impossible to fill your whole page of newsprint with stamps because there will always be gaps where the edges of the circles fail to meet up. These gaps are a big issue for nature, who likes to complexify (as opposed to complicate?) things by joining up smaller units without gaps.
Your knife meets the sphere and describes the plane that divides the grapefruit into two halves that are exactly equal to one another. If you slice it any other way, it wouldn’t make a great circle, you’d get plain old small circles anywhere but the middle. In order to make a great circle of a sphere, its radius has to be equal to the sphere radius.
The plane met the grapefruit and cut it into two equal hemispheres, exposing its great circle. The bright reddish pink flesh shows the “geodesic” of this sphere.
Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling in the gaps between them. In their study of solid-state crystalline matter, it is fundamental for chemists to consider how elemental particles pack together to form units. The most efficient way to pack spheres is the close-packed arrangement, which has two variants:
- Hexagonal close-packed
- Cubic close-packed
Thanks to http://2012books.lardbucket.org/books/principles-of-general-chemistry-v1.0/s16-02-the-arrangement-of-atoms-in-cr.html
Nature/God hates Gaps.
What’s so gauche about gaps?
According to nature, gaps are wasteful and inefficient. Natural forms want to fill space like there’s no tomorrow, and the way to do that is to get spheres nestled into an arrangement that ends up hexagon-ifying them to eliminate the gaps.
The image to the left shows the stamped geodesic surface (great circle) of the spherical grapefruit. You can compare that image to one below, which shows a different way of packing the circles into a perimeter in two dimensions. It is easy to see that one method of stacking the circle-stamps leaves less space free as gaps in between the stamps. Circle-packing has its own math that is important in understanding geodesics. If you pack the circles hexagonally, you get greater efficiency and that is what nature is all about. Just ask the bees, the bubbles, and the people at HexagonalAwareness.com. Circles settle up against each other the same way spheres do, forming straight lines.
Circles and spheres are wise to let the geodesic have its way with them, because its embrace confers:
- minimal energy – the fastest way from point A to point B is a straight line, even along the curve
- maximum stability – nestled hexagons support each other
- shape adaptability – geodesic membranes can be compliant and distribute forces efficiently instead of snapping or crumbling
Nature loves Hexagons.
So that’s the geodesic, bridging the gap between 2 and 3 dimensions, forming the hexagon and giving us the basis for a geodesic dome, which we will talk about in a moment. Let’s take pause and honour the absolute perfection of the sphere itself as a minimal-energy system unit.
How is it minimal-energy?
It all comes down to gravity. Think about a sphere as a very special polygon, one composed of an infinite number of faces of the same size, all being pulled evenly toward the centre of the sphere by its own gravity. At the same time, its molecular forces are pushing its boundaries outward in all directions from within. All the infinite points on the surface of the sphere are equidistant to the centre of the sphere. These equal forces of inward pull and outward push all spread out evenly across its surface to establish equilibrium. This endless stalemate makes the sphere the most efficient organisation of individual form in a primordial sense.
In general relativity, a geodesic marks the straight line an object takes relative to the curve of spacetime. When the grapefruit falls from the tree, it always follows a geodesic; in other words, it falls in a straight line from the tree to the ground.
The two packing styles listed above (hexagonal and cubic) are different mainly in how much free space they leave as gaps in a given volume and how they proliferate into volumes. Think of a fish tank full of ping pong balls that close-pack in arrays that give molecules and crystals their differentiating characteristics. You could say that the way that spheres organise themselves together in groups is fundamental to how our three-dimensional world creates itself.
It’s in the tissues
One particle, one cell, one droplet, one basketball, one planet – all spheres. But what about a creature, such as you or me, composed of quite a lot of cells, tubules, and other roundish structures? Let’s focus our growing understanding of geodesics on how nature tends to flatten the sides of round things in order to get them packing together neatly, without gaps. Geodesic tendencies appear in nature everywhere you look, including our cells and tissues. The hexagonal predilection isn’t limited to spheres and circles! It applies most emphatically to the tubules and fascicles of our fibres. Round things are very well-served by this configuration, and now you know why.
I just spent the morning superimposing hexagons on this traditional medical illustration I bought from Shutterstock awhile back. What I would really love for Christmas is an electron microscope so I can take some original photos of our inherent humanly hexagons!
Everyone loves Buckminster Fuller
Apologies for making you wait so long for Bucky. Richard Buckminster “Bucky” Fuller popularised the geodesic dome. He was an outspoken American author, architect, systems theorist, designer, and inventor. Thanks to PBS for the use of the picture above. Fuller published more than 30 books, and generated a polyhedron called the “Bucky Ball” that has become a legend in albiogenesis theories. He popularised such concepts as “Spaceship Earth” and synergetics, and in his long life was a much-admired revolutionary thinker. The origin of tensegrity has a scintillating history and owes its current relevance in the biomechanics world in large part to Bucky’s enthusiasm for the sticks-and-strings constructions that epitomise the concept.
A geodesic dome is derived by superimposing a network of geodesics (great circles) over the surface of a sphere or hemisphere. The great circles are flat-packed onto a 20-faced icosahedron, and as such their intersections form triangles that lend great structural stability. Note the triangular faces of the icosahedron form hexagons as the resolution scales up, and the hexagons retain their relations with the triangles. This is how the geodesics can approximated the sphere.
As one of the five Platonic solids, the icosahedron is one of the most hallowed of the polyhedra. There are many different kinds, 59 apparently, but the best-known is the Platonic, convex, regular icosahedron, whose 20 faces are subdivided into the triangles that approximate the original sphere. The greater the subdivisions, the closer to spherical the dome becomes, but it will always be a compromise of triangles. The 360 degrees we use to measure the circle is based on a sexagesimal (base six times ten) system, and as such Bucky used the 60 degree coordinate system of equilateral triangles in his theories.
Feast your eyes on the human embryo at the blastocyst stage and the geodesic leanings of our close-packed hexagonal nature.
“The opposite of nature is impossible.”
Thanks to everyone who contributes good things to the interwebs. As ever, bows of gratitude to Graham Scarr. I used these links: