Journal Pages
Learning:
The Journey of a Lifetime
or
A Cloud Chamber of the Mind

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Sunday April 15, 2007 6:40 am Edmonton

A. Morning Musings

6:40 am I do not have Internet access this morning.

B. Plan

Immediate    
Birds Birdwatching on the trip from Edmonton to Jasper 1 hr
Mathematics Continue reading "Algebra: Abstract and Concrete" by Frederick Goodman 1 hr
Literature

Read & make notes for book 4 of "The Brothers Karamazov" by Fyodor Dostoevsky

 1 hr
Later    
Technology Convert LPs to MP3 format  
  Make notes for chap. 4 of "Switching to the Mac"  
  Burn backup of images onto DVD  
Mathematics Read & make notes on The Humongous Book of Calculus Problems  
  Read "Symmetry" by David Wade  
  Make notes for "Mathematics: A Human Endeavor" ch 1  
  Read "Fearless Symmetry" chap 9: Elliptic Curves  
Model Trains Add ground cover to oil refinery diorama  
  Follow tutorial for version 8 of 3rd PlanIt  
  Continue assembly of coaling tower  
  Purchase DCC system  
Birds Create notebook pages birding in Mexico  
History Begin reading "Maya"  
  Read Watson "Ideas"  
Philosophy Read & make notes for "Breaking the Spell"  
  Begin reading "How Are We To Live?" by Peter Singer  
Literature New York Times easy crossword puzzles  
GO Complete reading "Lessons in the Fundamentals of Go"  
Puzzles

The Orange Puzzle Cube: puzzle #10
Major Goals    
Learning Review week's pages each Sunday  
  Review all pages for the month at the end of each month  
Technology Review & edit iPhoto files for 2006  
Model Trains Become proficient with 3rd PlanIt software  
  Install DCC on model train layout  
GO Learn to play GO at something better than a beginner level  
Drawing Learn to draw!! (I keep saying this, yet I have yet to put a pencil to paper).  
Mathematics Continue to play with mathematics.  
Literature Continue to read Literature  
Bird Watching Continue to engage in bird watching activities.  

C. Actual/Note

Mathematics 15

April 15

Mathematics Chronology

7:10 am Play time. I have cut out cardboard figures for both a square and an equilateral triangle.


Algebra: Abstract and Concrete. Edition 2.5 (online) Frederick Goodman

Chapter 1. Algebraic Themes

There are 12 sections to this chapter. The main themes appear to be symmetry (first four sections), permutations, divisibility, integers, group, rings, and fields.

1.1 What is Symmetry?

See Mathematics 12 notebook.

1.2 Symmetries of the Rectangle and the Square

See Mathematics 13 notebook.

1.3 Multiplication Tables

See Mathematics 14 notebook.

Before working on the exercises for this section I need to create a table of the positions of the vertices of the figure after each transformation.

For the square:

e
1 2 3 4
no motion
r
4 1 2 3
rotate p/2 clockwise
3 4 1 2
rotate p clockwise
2 3 4 1
rotate 3p/2 clockwise
a
2 1 4 3 .
rotate p along a axis
b
4 3 2 1.
rotate p along b axis
c
1 4 3 2.
rotate p along c axis
d
3 2 1 4.
rotate p along d axis

 

The following table means to perform the column motion first, then the row motion.

The row and column corresponding to e is easy, nothing changes. Similarly, two successive axis rotations returns the square to its original position.

Completing the rotations of r is equally straightforward, just add the exponents modulo 4.

e
a
b
c
d
e
e
r
a
b
c
d
r
e
.
.
.
.
e
r
.
.
.
.
e
r
.
.
.
.
a
a
.
.
.
e
.
.
.
b
b
.
.
.
.
e
.
.
c
c
.
.
.
.
.
e
.
d
d
.
.
.
.
.
.
e

However the remaining cells require a more careful matching of the motions with the final position of the vertices.

e
a
b
c
d
e
e
r
a
b
c
d
r
e
d
.
.
.
e
r
.
.
.
.
e
r
.
.
.
.
a
a
.
.
.
e
.
.
.
b
b
.
.
.
.
e
.
.
c
c
.
.
.
.
.
e
.
d
d
.
.
.
.
.
.
e

ar (that is, perform a motion of a and then a motion of r) gives 3 2 1 4. which is equivalent to d (using a table lookup of the transformations for the square).

The remaining cells are determined in a similar manner.
e
a
b
c
d
e
e
r
a
b
c
d
r
e
d
c
a
b
e
r
b
a
d
c
e
r
c
d
b
a
a
a
c
b
d
e
r
b
b
d
a
c
e
r
c
c
b
d
a
r
e
d
d
a
c
b
r
e

 

8:10 am I feel much better now that I have actually performed all of the transformations for the square. As Goodman suggests, creating a precise labeling for each of the motions is critical. So is the idea of noting which motion is performed first. A good way to begin the day.

 

8:00 PM I am typing this from a hotel room in Jasper. We had a very good birdwatching trip in from Edmonton. We saw three new lifers: American Wigeon, Barrow's Goldeneye in the park on the way in and a Hoary Redpoll in the townsite.

D. Reflection