Saturday, May 5
5:15 am
It is +7 °C, with a high forecast of +10 °C.
From the Environment Canada website:
Today Cloudy. Rain beginning this afternoon. Amount 5 mm. Wind northwest 30 km/h gusting to 50. High 10. UV index 3 or moderate.
Tonight Rain ending overnight then clearing. Amount 5 mm. Wind northwest 30 km/h gusting to 50. Low plus 2.
5:20 am Musings
I am feeling better this morning than I did yesterday morning. More awake, even though it is an hour earlier.
I have some work in front of me this morning as I slowly dismantle the new barbecue until I can dislodge the propane hose from the top control panel. But barring the unexpected I should be able to put it all back together again in less than an hour.
With rain forecast for later today I will size it up later this morning when it is light and decided whether to do the coulee walk or go in to the LFC and try walking on a treadmill for awhile.
I remember starting another online book on symmetry and modern algebra a few years ago.This might make a good third resource to consider as I recall it began with a description of the group of symmetry operations on a triangle and a square. Now to track that down. I think the author was Goodman.
6:20 am Mathematics
A search of the files on my computer using "Goodman" quickly identified a number of previous web pages that I had created in 2008. A check of a couple of these and I had the title of the book, "Algebra: Abstract and Concrete" by F. Goodman (2006). Success. Typing the title into google gave the website for the pdf file of the book. It is a free download (!) asking only that the user "make a generous donation to some organization devoted to relieving the misery of the world".
As an aside, I think this is the first time that I have used my website to find a previous reference.
I have even found the cardboard cutouts that I used to complete some of the early references.
Let me gather a few thoughts.
I now have 5 different resources at hand:
- Gilbert Strangs "Introduction to Linear Algebra" (2009), plus his MIT lectures on youtube
- Charles Pinter "A Book of Abstract Algebra" (1990)
- Avner Ash & Robert Gross (2006) Fearless Symmetry
- John Conway et al "The Symmetries of Things" (2008)
- Frederick Goodman (2011) "Algebra: Abstract and Concrete".
All 5 resources essentially cover similar topics, but with different emphases and examples. Let's try another conversation.
Me: Goodman's preface contains some important comments on the importance of thinking things through for yourself. He cautions that this is difficult and often frustrating and can't be learned quickly. It takes time. Which is what I have.
It has been 4 years since I last looked at the Goodman book, and I only made it through the first ten pages.
Goodman: "What is symmetry? Imagine some symmetric objects and some nonsymmetric objects. What makes a symmetric object symmetric? Are different symmetric objects symmetric in different ways?" [p. 1]
Me: These are good questions. It is easy enough to come up with a verbal answer but Goodman soon shows how a mathematician would answer this. A symmetry is a motion (i.e. transformation) that leave the object unchanged. For example, if one has a square and rotates it 90 degrees, then it still looks the same. An object is symmetric if it has symmetry motions.
This is the whole point of the Conway book! It then provides a notational system (first developed by Murray MacBeath) for describing these motions. This notation uses a star (*) and a point (.) where the star symbolizes the idea of series of mirror lines as in a kaleidoscope and the point symbolizes the idea that the symmetries fix a point. Thus *3. means 3 mirror lines of symmetry that fix a point in the center. An equilateral triangle has *3. symmetry. This quickly gets complicated but it serves a useful purpose when the patterns become complex.
I have now read the first three sections of chapter 1. They show the value of adopting some form of notation for representing different symmetries. The next section will show some connections between symmetry and matrix notation.
11:00 am Life
It is really miserable outside. The temperature has dropped to +3 C, there is a light rain, and the wind is about 40 kph.
The barbecue is assembled. It only took the removal of 2 screws and I was able to release the hose mechanism. The remaining steps proceeded smoothly.
Propane hose now hanging free
Electronic ignition system
Only the grills need placing
Assembly complete.
Now to clean up the mess.
The next step is to read the instructions on how to use it. Although it still used propane, the heating is via something called infrared.
[2:00 PM] The weather has continued to deteriorate through the day and it is now snowing. We made a mad dash to bring the flowers into the garage. The temperature is 0 C and -7 C with the wind chill factored in.
8:30 PM Literature
I have finished reading "Nikolski". Pure fun. A pleasure to read a book where the author enjoys himself.