#platonic solid

[muckmagister]

my cuubes:)

[knotty-et-al]

Tiny hexagon-spiral booklet with friend-shaped friend shapes of joy (platonic and archimedean solids drawn in isometric projection)

Front- and backside:

Laying at a stone to make the hexagon-spiral booklet appear more plastic:

This booklet can be folded into a tiny hexagon with moderate thickness:

[art-of-mathematics]

Today's Tiny Tetrahedron

(artist of the colorful artwork in the background of the first photo is unknown)

[travelling-my-little-pony]

Here is Fluttershy with a quartz platonic solid.

At the College of Psychic Studies, in London, England.

[curiosamathematica]

Going straight ahead on a complicated surface isn’t really that straightforward.

Mathematically, the paths you get by going straight ahead are called geodesics and are modelled by geodesic equations. Although they can be solved numerically and even explicitely for certain well-behaved surfaces (such as spheres), the evolution and long-term behaviour of these geodesics is very hard to predict.

A recent Numberphile video discusses a similar problem on polyhedra. These are piecewise linear, non-smooth surfaces, but again the behaviour of “going straight ahead”—simple as it may sound—can get really complicated. There have been new results on the dodecahedron, one of the Platonic solids, polyhedra that were thought to be well understood since antiquity!

Image by Toon Baeyens.

[hexeosis]

[geometrymatters]

Water Seems to Have Not One, But Two Distinct Structures in Its Liquid State

In 2018, scientists from the UK and Japan demonstrated that these peculiar properties have to do with the tetrahedral arrangement of water molecules in liquid form. This means that every water molecule is hydrogen-bonded to four others in a rough pyramid shape.

But still, how the structure is ordered has remained a topic of debate. One model proposes that water's molecular structure is unimodal - it's tetrahedrons all the way down. The other proposes that the structure is bimodal, consisting of two structures - tetrahedrons and something else.

To try and resolve the issue, industrial scientists from the University of Tokyo conducted computer simulations, and also ran experiments on liquid silica, one of the few other liquids known to have a tetrahedral molecular arrangement as well.

"We show the first clear numerical evidence in the structure factor for the dynamical coexistence of the two types of local structures ... supporting the two-state description of liquid water," the researchers wrote in their paper.

continue on sciencealert.com

Artilcle abstract:

Water is the essential liquid on earth since it not only plays vital roles in living systems but also has a significant impact on our daily life from various industrial applications to earth’s climate system. However, the unusual properties of liquid water, if compared with other liquids, has puzzled us for centuries because the basic structure of liquid water has remained unclear and has continued to be a matter of serious debate. Here, by computer simulations of three popular water models and the analysis of recent scattering experimental data, we show that there are two overlapped peaks hidden in the apparent “first diffraction peak” of the structure factor. One of them (ordinary peak) corresponds to the neighboring O–O distance as in ordinary liquids, and the other (anomalous peak) corresponds to a longer distance. We reveal that this anomalous peak arises from the most extended period of density wave associated with a tetrahedral water structure and is to be identified as the so-called first sharp diffraction peak that is commonly observed in silica and other tetrahedral liquids. In contrast, the ordinary peak arises from the density wave characteristic of local structures lacking tetrahedral symmetry. This finding unambiguously proves the coexistence of two types of local structures in liquid water. Our findings not only provide vital clues to settle a long-standing controversy on the water structure but also allow direct experimental access to the fraction of tetrahedral structures in liquid water.

https://pubs.acs.org/doi/10.1021/jacs.9b11211

[knotty-et-al]

Dodecahedron and stellated dodecahedron WIP - perspective drawing on isometric dot paper

So I drew this dodecahedron and then the stellated dodecahedron below.

And when I drew that stellated dodecahedron I made some very annoying mistakes (in the last steps!)...

As I could not erase those ink lines, I improved and cut out the parts with mistakes...

Sadly, the cut outs are irritating for the perspective drawing. (I will show you why shading the faces will help make sense of this shape once I have made a second drawing of it.)

Now I cannot shade the stellated dodecehedron as it would look very messy with the cuts.

So I just use the stellated octahedron part as an interesting collage item.

I could even insert neon-colored paper, which would make it look fancy.

But for making a shaded drawing I might draw a second one - but without those mistakes...