Monday, October 17, 2016
What shape is the square root of -1?
Yesterday we celebrated Quaternion day again, and the Hamilton Walk. Since it was Sunday, there were no school groups. There was a good turn-out of grown-ups, and quite few very small children, some in back-packs. The formula rescratching is always done by the youngest person present who understands algebra, and this year it was Caoimhin Malone, the youngest ever. He was willing to cede to Ferdia O'Cairbre, but F graciously declared that he had already done it, and was happy to let C try.
In view of the maturity of the group, I decided to open the spot quiz (before the song and carving, and usually restricted to children) to the whole assembly, and asked the question: What is the shape of the square root of -1?
A quaternion a +bi+cj+dk has a scalar part a and a vector part v=bi+cj+dk. We can consider v as a point in three-dimensional space, so if we have a set of pure vector quaternions, then it has a shape. In the real numbers, there are no square roots of -1. In the complex numbers, there are just two, so the set is snake-eyes. In the quaternions all the square roots are pure vectors, so what shape do they form? The prize was taken by Harun Siljak (or Shiljak -- that S has a v on top), who was first with the correct answer. Harun is the first known Bosnian participant in the Hamilton walk.
A letter from Don Chesley drew attention to a fine article by
Katharine Merow in the online Slate Magazine
http://www.slate.com/articles/health_and_science/science/2016/10/we_should_celebrate_hamilton_day_a_mathematical_holiday_on_oct_16.html
in which she proposes a global annual holiday on the 16th of October.
I suppose Harun is the same name as that of Haroun al-Rashid, that enlightened despot and patron of mathematics immortalized in the the tales of Sheherezade from the Thousand-and-one Nights. Reminds me of the trick mentioned by Des MacHale on Saturday, that involves taking a number such as 473473 or 287287, and dividing it in turn by "some random numbers, such as 7, 11 and 13." Des gave a brilliant acceptance speech when he was awarded the first ever Maths Week trophy to recognize achievements in popularizing mathematics. By common consent he was the outstanding choice.
Des passed on a very interesting question raised by a little boy in school: Where do the numbers go, when you rub them off the blackboard?
Sunday, June 05, 2016
Ben Bulben Hike
My friend and I walked up Ben Bulben on Friday. I've been up before, but thought it worth checking what information was on the web. Turns out that the info is misleading, so here is a heads-up if you are thinking of going there.
The nose of Ben Bulben, and the view back down.
I like to walk -- not climb -- up mountains, and get to the top.
When I google Ben Bulben hike, I get:
1. The Sligo tourism description of the "Ben Bulben Loop". This is a walk that doesn't go up the mountain at all, but lets you look at it from around the "nose". It involves about 45m of ascent. The summit of Ben Bulben is at 526m.
2. A number of descriptions and Youtube videos that describe climbs -- not walks. Some are described as "easy climbs". However, all these involve climbing up vertical bits of cliff. If you are not an experienced climber, they are good ways to lose your life.
3. Some descriptions of walks that involve clambering up rather steep slopes with loose stones on them. These are ok in good weather, provided you are careful.
I did not see a description of the straightforward way to walk up. This is steep, but not crazy. You take the public roads around the north side and up into the valley (Glendarragh) between Ben Bulben
aka Benbulbin) and Benwiskin. Park somewhere around the "scary bridge" (an 8-foot-wide slab of reinforced concrete without any parapet), grid G468703. You walk up by the well-used path that climbs southward up to the plateau. It runs up to the left of the river that runs down the southern end of the valley. Once you get up you can do a nice loop out west across the 500m summit and the neck to Ben Bulben (526m), back to the neck and then south-east to King's Mountain (462m) , and back north to the same path back down. Takes about 4 hours from the bridge. Bring a compass and the 1:50000 Sheet 16, in case you find yourself in mist. On Ben Bulben, be sure to keep on west to the nose, after you reach the summit. Stop when you get to top of the 45 degree slope, since a slip on that will take you over the cliff and end your life. The view all round and down from the nose is absolutely incomparable. If the weather is suitable, there is a path running close to the cliff edge on the north side, with spectacular views.
(The path up to the plateau is probably discussed on the web, but since I didn't run across it in a quick search, others might end up doing something they might regret, based on what they find online. Hence this post.)
The nose of Ben Bulben, and the view back down.
I like to walk -- not climb -- up mountains, and get to the top.
When I google Ben Bulben hike, I get:
1. The Sligo tourism description of the "Ben Bulben Loop". This is a walk that doesn't go up the mountain at all, but lets you look at it from around the "nose". It involves about 45m of ascent. The summit of Ben Bulben is at 526m.
2. A number of descriptions and Youtube videos that describe climbs -- not walks. Some are described as "easy climbs". However, all these involve climbing up vertical bits of cliff. If you are not an experienced climber, they are good ways to lose your life.
3. Some descriptions of walks that involve clambering up rather steep slopes with loose stones on them. These are ok in good weather, provided you are careful.
I did not see a description of the straightforward way to walk up. This is steep, but not crazy. You take the public roads around the north side and up into the valley (Glendarragh) between Ben Bulben
aka Benbulbin) and Benwiskin. Park somewhere around the "scary bridge" (an 8-foot-wide slab of reinforced concrete without any parapet), grid G468703. You walk up by the well-used path that climbs southward up to the plateau. It runs up to the left of the river that runs down the southern end of the valley. Once you get up you can do a nice loop out west across the 500m summit and the neck to Ben Bulben (526m), back to the neck and then south-east to King's Mountain (462m) , and back north to the same path back down. Takes about 4 hours from the bridge. Bring a compass and the 1:50000 Sheet 16, in case you find yourself in mist. On Ben Bulben, be sure to keep on west to the nose, after you reach the summit. Stop when you get to top of the 45 degree slope, since a slip on that will take you over the cliff and end your life. The view all round and down from the nose is absolutely incomparable. If the weather is suitable, there is a path running close to the cliff edge on the north side, with spectacular views.
(The path up to the plateau is probably discussed on the web, but since I didn't run across it in a quick search, others might end up doing something they might regret, based on what they find online. Hence this post.)
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