The Top 8

Since the blog has been going for a while, I thought I might celebrate some of the more popular posts and post series from the past years:

1. A series of posts about the 2013 World Solar Challenge:

2. A series of posts about the NetLogo simulation system:

3. Two posts about when and why Science began:

4. An explanation of Plimpton 322, a Babylonian mathematical clay tablet:

5. A post about the song The Wreck of the Old 97, and the story behind it – “a big day for Isaac Newton’s laws of motion but a bad day for Old 97 and its crew”:

6. A series of posts about chemistry in the kitchen:

7. A post about the reasons to study mathematics:

8. A series of posts about Science in Dante:

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Mathematics in action: Alea jacta est and Random graphs

Having posted about the mathematics of dice, here is a way to use dice to teach a seemingly unrelated concept, that of random graphs (also known as Erdős–Rényi networks).

Random graphs are networks where the links are introduced at random, among a pre-existing set of nodes. Here, the nodes will be the numbers from 1 to N = 20. The theory of random graphs, described in the classic textbook by Béla Bollobás, is quite beautiful.

The theory of random graphs is sometimes introduced as a game involving buttons and string. However, I cannot recommend that – it leads to the most horrible tangled mess. Instead, I suggest using a pair of icosahedral (d20) dice, like this one (such dice can be obtained from games shops, although I must confess to using random.org to do my dice rolls):

The first dice roll gives 2 and 4 (we re-roll any pairs which are doubles, or which have come up before). Each pair of numbers like this represents a random link in the network. We write the numbers 1 to 20 on a piece of paper, and draw a line between 2 and 4:

The second dice roll gives 1 and 12, so we add a link between those two numbers. A common measure of the density of links is the average degree, which is the average number of links per node. In this case, the average degree is the number of links divided by 10 (not by 20, because each link connects two nodes):

We keep repeating this process, and at an average degree of 1, something interesting starts to happen. A giant component begins to emerge – one big cluster containing many nodes (accompanied by smaller clusters and isolated nodes). The giant component is more impressive with a larger number of nodes, of course.

At an average degree of around log(N), taking (natural) logarithms to the base e = 2.7182818284…, enough random links have been introduced that the network becomes connected. We have log(20) = 3.00, but for this particular set of dice rolls, the network becomes connected at an average degree of 2.5. For larger networks, this connectivity threshold becomes quite sharp.

Additional random links reduce the average distance (mean path length) between pairs of nodes. This is the average number of “hops” needed to get from one node to another. Here the average degree is 6, and the average distance is 1.758:

For connected networks with an average degree of d, the average distance will be approximately log(N) / log(d). This function is shown by the red line below (with black dots showing measured values for our set of dice rolls). For large networks, this formula gives quite a small number, and the property of having small average distances is often called the “small world” property.

There is much more to the study of random graphs, of course, but we can see that a simple pair of icosahedral (d20) dice is enough to introduce the basics.

A Wellcome donation

The Wellcome Library has donated to Wikimedia Commons over 100,000 images relating to medical history, rare books, Asian art, and other topics. The images are available from wikimedia.org (progressively) or from wellcomeimages.org under a Creative Commons Attribution only CC BY 4.0 licence (giving credit to ‘Wellcome Library, London’). Example images from this treasure trove include:


Blow fly (Chrysomya chloropyga) – coloured drawing by Amedeo John Engel Terzi


Hebrew manuscript


Indian game of Snakes and Ladders


17th century Japanese herbal

Solar Impulse 2 prepares to fly

Solar Impulse 2 (photo above by solarracing.org) is a solar-powered aircraft with a wingspan of 71.9 metres, which will begin a planned around-the-world flight in late February or early March, starting from Abu Dhabi, where the Abu Dhabi Solar Challenge was recently held (and where the aircraft has recently arrived). The video below shows the aircraft’s maiden flight. It will be interesting to see how this goes!

Abu Dhabi Solar Challenge – wrap-up

The 2015 Abu Dhabi Solar Challenge is over. There was excellent coverage of this interesting event on Twitter, especially from solarracing.org. The race was also covered in local news, and some teams (like Michigan, Principia, and the Belgians) blogged about it. The map below shows where the various teams hailed from.

Although I previously estimated the final order, based on times for all three days, here is the order based on official results. Note that Tokai is now in 7th place.

2: University of Michigan Quantum (Twitter, Facebook, blog) 1st, official time 13:26:47
1: PI Solar Car Team (Twitter, Facebook) 2nd, official time 13:29:15
8: Punch Powertrain Solar Team (Twitter, Facebook, blog) 3rd, official time 14:10:02
34: Istanbul SOCRAT (Twitter, Facebook) 4th, official time 14:27:06
30: Team Arrow (Twitter, Facebook) 5th, official time 14:58:05
32: Principia Solar Car Team (Twitter, Facebook, blog, Picasa) 6th, official time 16:26:36
75: Tokai University Solar Car Team (Twitter, Facebook) 7th, official time 17:11:13
55: Hiroshima Institute of Technology 8th, official time 17:16:44
95: Apollo Solar Car Team (Facebook) 9th, official time 17:19:38
9: Team Onda Solare (Twitter) 10th, official time 17:53:25
15: ZHAW Solar Energy Racers (Twitter, Facebook) 11th, official time 20:42:33

The 2nd to 11th teams were behind Michigan by the number of minutes indicated: 1: 2.5, 8: 43, 34: 60, 30: 91, 32: 180, 75: 224, 55: 230, 95: 233, 9: 267, 15: 436. In graphical terms:

It is worth noting that Punch Powertrain (the Belgian team) were the fastest of the 4-wheel cars, and that the World Solar Challenge, in the Challenger Class and Cruiser Class, requires four wheels.

These last four teams all experienced trouble of various kinds (and therefore penalty time), but should be commended for participating:

256: Oregon State University (Twitter, Facebook) 12th, official time 27:11:29
17: Illinois State University (Facebook) 13th, official time 34:47:06
29: Eco Solar Breizh (Twitter, Facebook) 14th, official time 38:32:00
90: Team Okinawa Solar Car Project (Twitter, Facebook) 15th, official time 53:05:30

To finish my commentary, the picture below shows Michigan leading the way through the desert yesterday (photo: Noah Kaczor). Well done, Michigan!

This post has been updated with official times.

Abu Dhabi Solar Challenge Day 3

The route for day 3 of the 2015 Abu Dhabi Solar Challenge begins again at the Shams concentrating solar power station (photo below by solarracing.org), runs through historically important Liwa to the town of Ghayathi, and then returns to the power station. Racing on day 3 started early, at 8:00 AM local time.

The Shams power station is a 100 MW parabolic-trough solar facility, although it relies on a gas-operated booster heater to increase steam temperature from 380°C to 540°C. Even just with Google Earth, it looks quite impressive. Being subject to dust storms in its desert location, keeping the facility’s parabolic mirrors clean and shiny requires trucks with robotic scrubbing arms.

Live tracking (3D version here) allows fans from across the world to follow the Abu Dhabi Solar Challenge, which involves 15 teams, who finished the second day in this official order (official timings for the first two days shown in grey):

1: PI Solar Car Team (Twitter, Facebook) – 1st, 8:35:11
8: Punch Powertrain Solar Team (Twitter, Facebook, blog) – 2nd, 8:38:41
2: University of Michigan Quantum (Twitter, Facebook) – 3rd, 8:39:26
30: Team Arrow (Twitter, Facebook) – 4th, 8:59:02
34: Istanbul SOCRAT (Twitter, Facebook) – 5th, 9:04:03
32: Principia Solar Car Team (Twitter, Facebook) – 6th, 9:48:13
55: Hiroshima Institute of Technology – 7th, 10:39:19
95: Apollo Solar Car Team (Facebook) – 8th, 11:19:32
9: Team Onda Solare (Twitter) – 9th, 11:35:33
75: Tokai University Solar Car Team (Twitter, Facebook) – 10th, 11:55:29
15: ZHAW Solar Energy Racers (Twitter, Facebook) – 11th, 12:59:27
256: Oregon State University (Twitter, Facebook) – 12th, 13:30:05
17: Illinois State University (Facebook) – 13th, 21:01:48
29: Eco Solar Breizh (Twitter, Facebook) – 14th, 23:55:42
90: Team Okinawa Solar Car Project (Twitter, Facebook) – 15th, 31:49:48

As the map above shows, the current sequence of cars is 1, 2, 34, 8, 32, 55, 30, 75, 9, 95, 15, 29, 17, and 90, with 256 apparently getting a late start (the map also shows the dunes of the famous Rubʿ al-Khali to the south). Only minutes separate the lead cars, so today promises to be a real nail-biter.

Update #1: at 9:30 the sequence of cars was 1, 34, 8, 2, 55, 32, 30, 75, 95, 9, 15, 29, 17, 256, and 90.

Update #2: at 10:15 the sequence of cars was 34, 2, 1, 8, 55, 75, 32, 30, 95, 9, 15, 17, 90, 29, and 256, but with a continuing battle for 1st place. See the map below:

Update #3: at 11:10, on the way back to the start, the sequence of lead cars was 2, 1, 34, 8, 75, 30, 55. The clouds today seem to have made an impact.

Update #4: at 11:30, the sequence was 2, 1, 34, 8, 75, 55, 30, 32, 95, 9, 15, 17, 29, 256, and 90 – with a real battle going on among 7th, 8th, and 9th.

Update #5: the order of the first ten cars across the finish line was 2, 1, 34, 75, 8, 30, 95, 55, 9, 32.

Update #6: based on times for all three days, my estimated final order for the top ten is 2, 1, 8, 34, 30, 32, 55, 75, 95, 9. That may vary with penalties and protests from today.

Here are the cars that came first and second (photo by PI Solar Car team):