Mathematics in action: dice

We are all familiar with dice of various kinds. With fair dice, each number will come up with equal probability, regardless of how the die is rolled. This requires a degree of symmetry – we want a die to be a polyhedron where all the faces are equivalent. The obvious candidates are therefore the five Platonic solids, in which not only the faces, but also the edges and vertices are equivalent. The Platonic solids give us the common d4, d6, d8, d12, and d20 dice:

Five Platonic dice (d4, d6, d8, d12, d20) and two pentagonal trapezohedra (d10) – photo by “Copat”

However, the Platonic solids are more symmetrical than necessary for the job. A tetragonal disphenoid, for example, makes a very good d4:

A tetragonal disphenoid makes an alternative d4 – photo by “Traitor”

What is required is that a die be isohedral (also called “face-transitive”). Each face should be equivalent. Specifically, for any numbers A and B, given the die with A on top, there should be a series of rotations and reflections that make the die look like the starting position, but with B on top. This rules out shapes like the gyroelongated square bipyramid, where all the faces are equilateral triangles, but the triangles are not equivalent (the “end” triangles differ from the “middle” triangles):

A gyroelongated square bipyramid does not make a fair die – photo by Andrew Kepert

We also want a die to be convex, so that it can land on its faces. Stellated polyhedra are excluded:

Stellated polyhedra cannot be dice – mosaic in St Mark’s Basilica, Venice

Trapezohedra satisfy this “convex and isohedral” rule, and the pentagonal trapezohedron is commonly used as a d10 (see the picture above). Trapezohedra work best with 10, 14, 18, … sides, since then pairs of faces can be parallel, and there can be an unambiguous “top” number. The cube can be seen as a special case of a trapezohedron.

For 12, 16, 20, …. sides, bipyramids make good dice (the octahedron is a special case of a bipyramid):

A bipyramid makes a good d16 – photo by “Traitor”

These are not the only shapes satisfying the definition, however. The 13 Catalan solids also satisfy it, and some of them make good candidates for dice. For example, the deltoidal icositetrahedron and the tetrakis hexahedron are both good candidates for d24:

The deltoidal icositetrahedron and tetrakis hexahedron are alternatives for d24 – photos by Jacqueline de Swart (left) and “Traitor” (right)

Some Catalan solids, like the pentagonal icositetrahedron, are unsuitable as dice because there is no unambiguous “top” number. On the other hand, there are some additional variations that are isohedral, like the hexakis tetrahedron.

For more on this subject, see Alea Kybos’ impressive dice page.

Noctilucent clouds!

The beautiful Noctilucent clouds (photo above by Kevin Cho) are slightly misnamed. The name means “night-shining clouds,” but in fact they are only visible between the end of civil twilight and the end of astronomical twilight (and at latitudes north of 45°N or south of 45°S).

Noctilucent clouds are the visible form of polar mesospheric clouds, made up out of ice crystals in the normally very dry almost-vacuum at extremely high altitudes, around 80–90 km up (see NASA image of a polar mesospheric cloud above).

Noctilucent clouds were first identified as a distinct atmospheric phenomenon in 1885 (see the Google ngram below for uses of the phrase). The ice crystals from which noctilucent clouds are formed require both water vapour and dust for crystal growth nuclei. The sources of these ingredients is still mysterious, although at least some of the dust may come from meteors.

Yet another atmospheric phenomenon for the bucket list!

Scotland the Brave


Scotland is voting today on whether, after 307 years, to leave the United Kingdom and follow the uncertain path of independence. It seems a good moment to salute Scottish scientists. Three (of many) are shown above:

Update: it seems that Scotland has voted to remain as part of the UK.

Mohr’s burette

The burette, an important volumetric tool, was invented by Karl Friedrich Mohr somewhere around the middle of the 19th century. There had been other devices carrying the name “burette,” but they required pouring. Mohr introduced a rubber tube with a clamp that allowed the gradual drop-by-drop flow needed for titration (the clamp was later replaced by a tap). Mohr’s 1855 book on titration, which illustrated the device, helped it to become the key item of analytical equipment it still (in spite of more modern digital devices) is today. Thanks, Karl!

Once again, a Google ngram summarises the history, with the word “burette” rapidly gaining popularity from 1855, but being replaced by the word for the process, which itself faded after 1960 – perhaps because of the growth of other kinds of science.

Emission spectrum scarves

Above is one of the nifty atomic emission spectrum scarves made by Becky Stern. An emission spectrum is the pattern of light colours produced by heated atoms, and the emission spectra of different chemical elements act as a kind of visual “fingerprint.” There’s something wonderfully geeky about putting one on a scarf – and they look great too.

Becky Stern no longer seems to sell these scarves, but has posted patterns for making your own. More pictures here.

Dendrogramma, a strange new genus

A recent paper in PLoS ONE (by Jean Just, Reinhardt Møbjerg Kristensen, and Jørgen Olesen) reports the new species Dendrogramma discoides (marked with asterisks in the image from the paper above) and Dendrogramma enigmatica (unmarked above). These species were found in waters off south-east Australia.

Specimens from this new genus were collected in 1986, and preserved in formaldehyde. This is a pity, because the organisms have not been found again, and the formaldehyde has destroyed their DNA.

DNA analysis would have been useful to work out in which animal phylum to place Dendrogramma. The usual candidates (left to right above) are Porifera (sponges), Ctenophora (comb jellies), Cnidaria (jellyfish), and the Bilateria – such as Echinodermata (which are bilaterally symmetrical as larvae) or Chordata.

The fascinating Dendrogramma specimens are not bilaterally symmetrical, and are not sponges. They lack specialised features of the Ctenophora and Cnidaria, such as Cnidarian stinging cells. So what are they? It seems likely that either a new phylum has to be defined; or that Dendrogramma must be placed in a restored Coelenterata (a former phylum which once contained the Ctenophora and Cnidaria); or that the boundaries of Ctenophora or Cnidaria need to be extended – but without DNA, the decision is difficult.

Ebola #3

Updating Ebola news, the map above shows the current situation in West Africa. The graph below shows cases to date (with LOESS smoothing).

Below is an estimate of the new cases per day (calculated from the smoothed data). The acceleration of the disease appears to be continuing, which is very disturbing. See the WHO and CDC websites for more information.