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:

• Robert Brown (discovered Brownian motion): born Montrose, 1773
• James Clerk Maxwell (the great physicist): born Edinburgh, 1831
• Alexander Fleming (discovered lysozyme and penicillin): born Lochfield, 1881

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 19^th 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.

Bárðarbunga erupts!

The Bárðarbunga volcano in Iceland has been erupting in a spectacular way. The photograph below was taken by Peter Hartree on 4 September (click to zoom):

Spring!

It’s Spring here in the Southern Hemisphere!

Mathematics in action: check digits and ISBNs

International Standard Book Numbers or ISBNs (the older, 10-digit kind, not the newer 13-digit kind starting with 978) have an interesting way of detecting errors made in writing down the numbers. The number at the top of the image above is an example of what I mean. The process works like this:

1. Circle the right-hand digit. This is the check digit. If it’s an “X,” that means 10. In the image above, the check digit (5) is circled in red.
2. Copy the other 9 digits, and write 10, 9, 8, 7, 6, 5, 4, 3, 2 underneath them.
3. Multiply vertical pairs of numbers, and add up the products. Call this S. For the example above, S = 0 + 81 + 24 + 42 + 18 + 40 + 20 + 12 + 0 = 237.
4. Divide by 11, and call the remainder R. For the example above, R = 6.
5. The check digit should be 0 if R = 0, and 11 − R otherwise. If not, something is wrong. For the example above, the check digit is 11 − 6 = 5 (for students familiar with modular arithmetic, we can express this more simply: the check digit should be −S mod 11).

This scheme will detect any copying error where a single digit gets changed. Imagine that one of the digits is A (ranging from 0 to 9), in the position where it gets multiplied by N (ranging from 10 to 2), and A gets replaced by a different digit B (ranging from 0 to 9). The sum S will increase by (B − A)N, which cannot be a multiple of 11. Therefore the remainder R will change, and the check digit calculated in step 5 will no longer match the circled one.

The ISBN system also spots digits being swapped. Imagine that one of the digits is A (ranging from 0 to 9), in the position where it gets multiplied by N (ranging from 10 to 2). Another digit is B (ranging from 0 to 9, with B ≠ A), in the position where it gets multiplied by M (ranging from 10 to 2, with M ≠ N). If the digits A and B are swapped, by a similar argument to the one above, the check digit calculated in step 5 will no longer match the circled one.

A number of similar schemes exist, since the problem of people writing down numbers incorrectly is widespread, and detecting such errors is often worthwhile. Australian Medicare cards, like the one above, operate as follows:

1. Ignore the right-hand digit, which is a sequence number. Circle the digit second from the right. This is the check digit. In the image above, the check digit (8) is circled in red.
2. Copy the other 8 digits, and write 1, 3, 7, 9, 1, 3, 7, 9 underneath them.
3. Multiply vertical pairs of numbers, and add up the products. Call this S. For the example above, S = 1 + 6 + 21 + 36 + 5 + 18 + 49 + 72 = 208.
4. Divide by 10, and call the remainder R. For the example above, R = 8.
5. The check digit should R. For the example above, the check digit is 8.

This scheme will also detect any copying error where a single digit gets changed. Imagine that one of the digits is A (ranging from 0 to 9), in the position where it gets multiplied by N (1, 3, 7, or 9), and A gets replaced by a different digit B (ranging from 0 to 9). The sum S will increase by (B − A)N, which cannot be a multiple of 10 (this is why 5 is excluded as one of the multiplying numbers). Therefore the remainder R will change, and the check digit calculated in step 5 will no longer match the circled one.

However, the Australian Medicare Number scheme only detects some digit swaps. Even if we only worry about swapping adjacent digits, it does not catch all such errors. For example, 127456786 and 172456786 are both valid. If you imagine that one of the digits is A (in the position where it gets multiplied by N) and the adjacent digit is B (in the position where it gets multiplied by M), you can see that sometimes the change in S from a swap will be a multiple of 10, leaving the check digit unchanged.