Watching The Cycle, an Iranian movie tracing the hospital blood supply back to those desperate enough to sell blood, the geeks of that society, geek in the old sense of chicken head carnival act, got me thinking about blood flows more generally.
Terry has lent me a copy of Shape and Structure: From Engineering to Nature, by Adrian Bejan. The book in reminiscent of a D’Arcy Thompson work, in focusing on how form follows function.
You might find where I was querying Perplexity about the relevance of Bejan to the GST approach to eco-thermodynamics, i.e. surfing the solar gradient per Into the Cool, another book on the ISEPP shelf.
The Cycle is a 1977 film set and made in Iran, that won much acclaim on the film festival circuit.
Bejan cites Geoffrey West, author of Scale, a book here in my own collection. Scale is all about power relationships by which I don’t mean in geopolitics but in geometry. When you grow or shrink a shape by a scale factor, the change in linear dimensions is a 3rd root of the change in volume. If I double all the edges of a cube, its volume multiplies by a factor of eight i.e. two to the 3rd power.
How is a heart supposed to pump oxygen to every nook and cranny in a camel or elephant, dinosaur or giraffe, when there’s so much volume? The answer of course is tubing, and what Bejan shows is that two trees intertwined, one sending warm blood distally, the other harvesting cooler blood to bring back, actually reinforce each other in measurable ways, bumping that 3rd to 2nd root transition, from size to body surface area, to a 4/3rd root instead (still a 2/3rd root if cold-blooded). Klieber’s Law.
In grade school we might look at it this way: we know the D-edged tetrahedron has about 6% less volume than its R-edged cubic counterpart, where D = 2R, diameter and radius respectively, of our reference sphere. S3 is the signifier we sometimes use, for the R-cube/D-tet ratio of 1.06066, the 3rd root of which we call S1. Take that D-edged tet, say made of clay, and add to any face, by a linear amount of S1 times initial length, which isn’t much. Surface area will increase by a factor of S2. S1 = ~1.01982445133,
One of my Google slide deck slides cites
Huffington Post as my source for a D’Arcy Thompson letter, written to Alfred North Whitehead. He wants to know where all this “three dimensional” jazz is stemming from, in the process of questioning his own allegiance to orthogonality. His concerns prefigure a later architect’s. Can’t a tetrahedron bear the brunt, when we talk quaintly about
the square-cube law?
A cool thing about a tetrahedron is: if you slice parallel to any face, like from a tetrahedral loaf of bread, or chunk of cheese, all six edges will decrease in proportion. The face has been foreshortened, and so have been the edges to the opposite corner, keeping all the angles the same overall, including central angles, meaning no change in shape, only size.
In contrast, a clay brick, cut parallel to any face, will keep its 90 degree surface angles yet suffer changes in relative proportions, meaning in radial central angles. A brick has to shrink inwards in all directions, giving that more tesseract-looking outline, of cube within cube. The tetrahedron keeps shrinking towards a corner.
This distinction between angles (central and surface), meaning shape, and scale (meaning size), is commonly considered core curriculum. The size spectrum is likewise our frequency spectrum, from the tiniest intervals to the largest we need. This might as well be the electromagnetic spectrum of polarized oscillations, otherwise known as vibrations, or frequencies. Pure shape = Platonic.
Our beginning might be methane, or CH4, with the idea of an idealized gas per Avogadro, a uniform distribution of molecules, but with random variations from CCP origins. Then zoom in on one molecule and focus in on the angles of 109.47 degrees between any two CH vectors. Arccos(-1/3). We learned trig for a reason apparently.
We know have a fun API for the XYZ apparatus, with four basis vectors instead of three primary and three secondary. No differences in class, between positives and negatives. All twelve permutations of (2, 1, 1, 0) will give the 12 radials of a home base shape, the one we associate with dynamic equilibrium. Chaotic perturbations test our sanity sometimes.
