Mathematical Models

Way back when I was at school, I came across a fascinating little book called Mathematical Models, by Cundy and Rollett.  It had instructions on how to build various mathematical objects, such as stellated polyhedra.  I liked the book so much I actually bought my own copy, new, for £2.95, which was a lot of money back then!

I made a few of the simpler models, but never got much further than the dodecahedron.  I certainly never got as far as making any of the fiddly stellated ones.

So today, when I came across a posting in Google+ about Anselm Levskaya's website polyHédronisme, I was taken right back to those days.  Playing with this interactive web-based systems is much easier than fiddling with card, glue, and scissors, though.  Type in a few commands, and a zoomable, rotatable polyhedron appears!

I've spent my afternoon playing around on this site, and reading up on Conway polyhedron notation that is used to define shapes, and now I can say I have at last "made" some of these polyhedra.

The small stellated dodecahedron is made by raising a pentagonal-based pyramid on every face of a regular dodecahedron. If the pentagons making up the dodecahedron have side length \(1\), then the height of each pyramid should be* \[ \frac{\sqrt{4\sqrt{5}-1}}{2} \approx 1.41\] The Conway notation command in polyHédronisme that achieves this is \(k(5,1.41)D\), which means: start with a dodecahedron \(D\), then raise a pyramid of height \(1.41\) on each \(5\)-sided face.

The great dodecahedron is made by making a pyramidal dimple in every face of a regular icosahedron. If the triangles making up the icosahedron have side length \(1\), then the depth of each pyramid should be* \[ \sqrt{\frac{1}{2}- \frac{\sqrt{5}}{6}} \approx 0.36\] The Conway notation command in polyHédronisme that achieves this is \(k(3,-0.36)I\), which means: start with an icosahedron \(I\), then indent a pyramid of height \(0.36\) on each \(3\)-sided face.

The great stellated dodecahedron is made by raising a pyramid on every face of a regular icosahedron. If the triangles making up the icosahedron have side length \(1\), then the height of each pyramid should be* \[ \sqrt{\frac{7+3\sqrt{5}}{6}} \approx1.51\] The Conway notation command in polyHédronisme that achieves this is \(k(3,1.51)I\), which means: start with an icosahedron \(I\), then raise a pyramid of height \(1.51\) on each \(3\)-sided face.

So much for standard polyhedra.  But polyHédronisme doesn't stop there.  I had great fun playing about with the notation language, producing weird and wonderful forms:

(i)  \(k(20,1)bk(3,2)I\)    (ii)  \(k(12,1)k(10,2)bk(5,1)D\)    (iii)  \(k(20,-0.3)k(6,0.3)bk(3,-0.3)I\)
(iv)  \(k(24,-0.5)k(6,0.2)k(20,-1)bk(3,-0.1)k(5,1)D\)

The results of play can only really be appreciated on the site itself, rotating the polyhedra, and getting a real feel for their shapes, with all their dips and bumps.  A marvellous site.

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* The book Polyhedron Models has helpful stellation diagrams that allow these heights to be calculated, with a little trigonometry.

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Ctrl-Shift-T

We've gone over to using Google products at work.  For the most part, I like Google Mail.  It has its quirks, but the selling point for me is that, being browser based, mail items have URLs, so I can just paste the relevant URL into my to-do list or Evernote daybook, rather than clutter up my inbox with emails I'm waiting to action or need for meetings.

However, Gmail has a user interface feature I just keep getting wrong.  I consistently click the "Compose" button to move to my contacts, instead of clicking the "Mail" menu, and selecting from there.

I keep clicking "Compose", instead of selecting "Contacts".  Why?

I don't know why I keep doing this (maybe because the first two letters are the same?), but I do.  All the time.  I don't blame myself: I have been well-enough educated in user interface design to believe that if the user consistently makes a mistake like this, the UI is to blame.

This particular mistake is not that serious: I just have to keep binning unwanted empty emails (and muttering imprecations at the interface).  But there's another mistake I keep making, this time with the Chrome browser: I often hit "close tab" instead of "back" when I finish reading a linked page.  And then I've lost that tab's history, in particular, the page I was trying to return to when I closed the tab.

But then the other day, I discovered that all is not lost!  Hit Ctrl-Shift-T, and the tab is automagically resurrected.  Hit it again, and the previously closed tab reappears. And so on. Yes!  A key combination to undo a user interface infelicity.

So why isn't this option in the menu?

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a normal April again

Last April was anomalously wet.  This year, things were back to "normal", rainfall-wise, if not temperature-wise.  Despite being much drier, 2013 is still the second wettest April since (our) records began, with a measured 14mm:

April rainfall, in mm

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How far to the Crowne Plaza?

We've just reserved our hotel room for Eastercon 2014.  The con is being held in the Crowne Plaza Glasgow.  The online booking system helpfully gives an indication of how far the various con hotels are from the venue.

One property of a metric is that the distance from somewhere to itself, d(x,x) = 0, for any x. However, the booking system seems to be using a more science fictional metric, since it thinks that d(Crowne Plaza, Crowne Plaza) = 176 yards.

Possibly the hotel is beside itself with joy at hosting Eastercon?

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pond life

Our pond "water feature" has stopped working. That probably means the filters are clogged. Getting to the pump to unclog it is a bit of a palaver, however.

(left) that's one small step for a man; (centre) wading out to the deeps; (right) you're not taking photos for your blog, are you?

And yes, the filters are very clogged.

(top) cleaned filter shining bright; (bottom) clogged stone and foam filters, still to be cleaned

After a second wading expedition to replace the pump, all is now right with the water feature.

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Spring, beautiful, but needs work

The apple tree is blossoming.

But to keep a garden in flower requires work.  Here's our to do list for today:

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farewell, Jedi Day

All eleven ducklings returned to roost tonight.

10 in a flock, plus an outlier

The photo was taken at about 8:40pm BST, as it was getting dark.  So the shot has been brightened, to make it light enough to see the ducklings, which is why it is a bit grainy.

There's a slight pinkish tinge to the water.

That's because the sky above looked like this:

click to give that full screen effect

This shot has not been shopped at all (apart from being cropped).  The sky really did look like that!

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