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?