Wednesday April 2, 2008 5:30 am Lethbridge, Alberta

This page last updated on: Thursday, May 1, 2008 6:46 AM

It is - C with a high forecast of +5 C. Sunrise 7:05 Sunset 20:04 Hours of daylight: 12:59

A. Morning Musings

I like my routine with the mathematics now part of the mix. And I like my early morning coffee.

coffee

Learning Category	Planned Activities for Today	Time
Literature	Begin morning with a Rumi reading
Literature	Continue reading "Time Regained" by Marcel Proust	2 hr
Mathematics	Continue reading "Fearless Symmetry" by Ash & Gross	1 hr
Mathematics	Continue reading "Symmetry" by du Sautoy	2 hr
Mathematics	Continue reading "Symmetry and the Monster" by Ronan	1 hr

B. Actual Learning Activities

6:00 am

Notes on Symmetry - 3

Dale Burnett

From my notes of Symmetry - 1:

du Sautoy also talks about always having a yellow notepad at his disposal. This reinforces my idea of a lined notepad of high quality paper and a good fountain pen. My notepads have a bright orange cover. My fountain pen is a Pelikan M250 Tradition with a fine nib.

1. Use my camera to take photos of objects that appear to possess symmetry. Begin to create a catalogue of such images.

2. Begin making notes using my fountain pen and Rhodia notepads.

From my notes of Symmetry - 2:

I am going to see if I can create a chart to monitor my reading of books on symmetry. Each cell will correspond to a chapter. Yellow indicates the number of chapters in the book, green indicates that I have read and made notes on the chapter.

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
Symmetry
Fearless Symmetry
Algebra
Abstract Algebra
Creating Escher-type Drawings
Handbook of Regular Patterns
Symmetry & the Monster

M. du Sautoy. (2008). Symmetry.

Chapter 2 September: The Next Roll of the Dice

My approach for a serious book like this is to begin by simply reading the next chapter with a pencil in my hand. When I encounter something I like, I make a small mark in the margin. If the chapter happens to contain some activities or exercises then I will decide which of these I want to try. I will do these with pen & paper. Then when I have completed the chapter I go back and try to make a few notes for this web site. This will include a scan of the work I did with pen & paper.

Now to read.

...

Now to make a few notes.

"September for me has always been a month of beginnings. Ahead lies the cycle of the academic year with all its promise of new things to learn and discover." [p. 32]

True enough. Also January.

Now that I am retired, the rhythm of the academic year is no longer present. On the other hand that frees up my sense of time. I can now seize the moment with much more freedom than I had before.

"I like to work from home. I find an office an oppressive space, constantly reminding me that I'm not having any great ideas." [p. 32-3]

For me, the office was a social place. This was where students and colleagues could meet to discuss course related matters. Even with colleagues, the interaction was primarily about the program rather than ideas. Ideas need quiet. The trick is to find the proper balance between the two environments.

"My room at home is a space where I can easily drift in and out of ideas without feeling guilty. It is an extremely messy place. ... But it is actually a good reflection of my thought process. I'll start looking for a book buried deep in the pyramids of paper that sit atop my desk. But during the search I'll often come across something I hadn't been looking for which can take my thought process off in an unexpected direction. By maintaining an untidy room I'm raising the likelihood of making these random connections. Whenever I tidy everything up, the potential for getting random ideas gets filed away too." [p. 33]

Yes, but ... there is also value in being able to find a needed book in a hurry while one is still having the original thought. The trick here is to have both balance between mess and order and to have lots of books.

"The other stimulant in addition to music that I use (or abuse) at home comes from my espresso machine. The ritual of coffee making forms an important skeleton around which to frame my working day." [p. 33-4]

This sounds familiar. But I add the ritual of adding a page to my web site.

"So, a 15-sided regular polygon has 15 rotational symmetries. But now something interesting happens. The symmetries of the 15-sided figure are actually built from the symmetries of two smaller shapes: the pentagon and the triangle. If I draw a pentagon and a triangle inside the 15-sided figure, I can achieve every rotation of the larger shape by combining rotations of the smaller shapes." [p. 34]

I did not know this.

du Sautoy does a great job of stimulating my interest. I can now see value in having a collection of cardboard shapes near at hand.

This raises the question of what shapes? Triangles for a start. Definitely equilateral. Now consider squares. Two possibilites come to mind: squares with a side equal to the length of the equilateral triangle, and squares with the diagonal equal to the side of the equilateral triangle. I also now want a triangle that is formed by cutting the square into two along a diagonal. And I want quite a few of each shape so that I can tile a small area. And, I think I want each shape to have a distinct color so I can see the patterns of such a tiling. I would like to extend this to all polygons of at least 12 sides.

How many different shapes can I create by simply cutting an equilateral triangle along its 3 axes of symmetry? What are the lengths of the sides? Suppose the original equilateral triangle has a side of length 1? 2? 3? 4? Suppose the original length is a prime number, or not a prime number?

This is beginning to explode. Fantastic. Imagine a group of 20 12 year old kids playing with this idea for a couple of weeks. Imagine them using the Web to find out more about what they are doing. Imagine this being written up for a math journal or a math conference.

This has something to do with architecture as well ...

Surely there are packages of such shapes in educational/mathematical shops. I wonder...

"You cannot break the pentagon or the triangle into rotations of smaller shapes, as you can with the 15-sided shape. The reason is that 5 and 3 are both prime numbers. ... If you take a regular two-dimensional polygon with a prime number of sides, then the rotational symmetries of this prime-sided shape cannot be built from those of smaller symmetrical objects. Not only that, but these prime-sided figures are the building blocks for the symmetries of all the other regular two-dimensional polygons. ... This is a geometric way of saying that every number is built by multiplying primes together" [p. 35]

This is related to the Fundamental Theorem of Arithmetic.

Googling this gives more information.

"As I discovered during my training at university, turning space into numbers is the most powerful language for describing higher-dimensional objects." [p. 38]

This reminds me of my first exposure to a computing language called APL. I quickly saw how to create a 2-dimensional array of numbers and then wondered what would happen if I tried to create a 3-dimensional array. I thought this might generate an error message, but instead it created a printout of a number of 2-dimensional arrays which were clearly slices of the cube of numbers. I then tried to create a 4-dimensional array. Much to my surprise, this also worked. It printed out a series of cubes of numbers, each of which was a 'slice' of the 4-dimensional cube. It was clear that this idea would could be continued to the limits of the computer. I have never forgotten that afternoon. I can even remember what the room looked like.

"It's the weekend. ... a morning in the British Museum looking for symmetry."

I want to pursue this idea with my digital camera.

I once had the idea of taking photos of the buildings and fences at Byron Bay and making a book of these images. Perhaps I should resurrect this idea, but add the theme of symmetry. Hmmm....

This has been another exciting mathematical morning.

Tags: mathematics, symmetry

4:20 PM

I am in a departure area in the Calgary airport waiting for the flight to Vancouver which is due to leave at 6:10 PM. I can pick up a wireless signal, but I do not have a telus hotspot account so I will give that a pass.

I am having a very relaxing trip and will sit back and read a bit more Proust before boarding the plane.

Books on the Go Today


	see below

Sautoy

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Wednesday April 2, 2008 5:30 am Lethbridge, Alberta

A. Morning Musings

B. Actual Learning Activities

6:00 am