Euclid Book XIII Proposition 18

On Aristotelian physics, I was wondering about the last line of this Proposition: To set out the sides of the five figures and compare them with one another. Here Euclid proves that if a dodecahedron and an icosahedron are inscribed in the same sphere, then side of the icosahedron is greater than the side of the dodecahedron. One might think (because I did) that the icosahedron, having twenty faces, would have a greater volume than the dodecahedron inscribed in the same sphere, because the dodecahedron has only twelve faces and so should be a poorer approximation to a sphere as a result, but this intuition is false. One similarly might expect the sides of the icosahedron to be shorter than those of the dodecahedron for similar reasons, but neither is that true. However, the dodecahedron and icosahedron are dual each to the other, meaning that they each have the same number of edges, and if you swap faces with vertices you get the dual polyhedron. In both figures, opposite edges are parallel, so that you can turn one into the other just by rotating their edges through a right angle. These images were all stolen from George Hart. 

Clearly, if the two figures are inscribed in the same sphere this will not be possible because the side lengths will not match, but if they each turn into the other then their volumes and surface areas will also be exchanged. According to Wikipedia, "Apollonius of Perga discovered the curious result that the ratio of volumes of these two shapes [circumscribed by the same sphere] is the same as the ratio of their surface areas". An icosahedron sits inside a cube, with six of its edges lying in the faces of the cube, as you can see in George Hart's virtual polyhedra museum. There are five ways you can do this, so the icosahedron is contained within a compound of five cubes. 

The parallel edges of the icosahedron are each opposite sides of a Golden Rectangle, the diagonal half of which is the right-angled triangle constructed in Proposition 10 of Book XIII of Euclid's Elements. 

The same duality holds for the cube and the octahedron:

Because of this, an icosahedron can be constructed by joining the midpoints of the edges of an octahedron:

The tetrahedron is self-dual in this sense, and the tetrahedron and its self-dual both sit within a cube.

You can also fit a cube inside a dodecahedron and this too can be done in five ways

Hence the rate of exchange is five beds for one house. See A New Kind of Science. 

Now in Aristotelian physics, the behaviour of bodies such as that of water and air is due to their different shapes. Air being octahedral, water being icosahedral, earth being cubical and fire being tetrahedral.

So I started to wonder about the behaviour of mixtures of water and spirit and I found these two demonstrations:

 

See All About Water and Entropy and Documentary About Water for more on how water itself has to different molecular states.

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And when this salt trick worked, I could almost imagine I saw the volume of the liquid in the bottle expand just a tiny bit! 

You can also make water rise by heating it, but you would need to keep the hot water contained in a bag, like an underwater hot-air balloon:

See Mechanical Equivalent of Heat. 

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For more Aristotelian material science, see More History of London.