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

February 2006 Mathematics Notebook

An Example of a "Learning Process" Journal

The intent is to continue making notes for Coincidences, Chaos, and All That Math Jazz. I read and yellow highlighted Part III "Exploring Aesthetics" yesterday afternoon. It is about visualization and imagery in mathematics and in life.

Chapter 7 From Precise Beauty to Pure Chaos.

• The Golden Ratio is .
• A Golden Rectangle is a rectangle for which the ratio between its base and height is the Golden Ratio.

• When we remove the largest square from a Golden Rectangle, we obtain another Golden Rectangle. This process can be repeated forever.

The reverse process should also be clear in my imagination. I should be able to take a Golden Rectangle and build another one by constructing a square equal to the size of the largest side and then add on a piece using the Golden Triangle. Once again, this should be crystal clear in my mind. It is now, but I must be able to see this when I view these notes at a later time.

• "In fact, the Golden Rectangle is the only rectangle that possess the property that after the largest square is removed, the remaining rectangular offspring has the same proportions of length to width as the original parent rectangle has." [p. 137]

• A Golden Triangle is a right angle triangle with sides 1, 2 and the Golden Ratio.
  
  
• "Given any triangle, four copies can be arranged to make a similarly proprtioned triangle." [p. 141]
  
  
• "The Golden Triangle is the only right triangle for which five copies can be assembled to produce a larger copy of itself." [p. 142]
  
  
• "We can tile the entire plane by repeatedly building super-Golden-Triangular tiles. ... When we do so, the tiling that results is surprisingly chaotic or jumbled-looking. ... It can be shown that there is no translation of this tiling that will result in that shifted version matching up perfectly with the original tiling." [p. 143 - 145].

Chapter 8 Origami for the Origamically Challenged

• The fractal image known as the "Dragon Curve" can tile a floor with no gaps or overlaps. [p. 147]

• The chapter begins with a simple paper folding exercise and noting the pattern of the folds. This leads to a formula for generating the pattern. This leads to a second, totally different way of generating the pattern. Finally, a discussion of Turing machines leads to a simple five-line Turing Machine program that also generates the pattern.
  
  
• "A fractal is any geometric object that has infinite complexity." [p. 163]

Chapter 9 A Twisted Turn in an Amorphous Universe

• "The moving power of mathematical invention is not reasoning but imagination. - Augustus de Morgan" [p. 166]
  
  
• "Our goal is to develop agility in our thinking." [p. 167]
  
  
• "... a recurring theme throughout the book - how we think and how we make sense of the world." [p. 179]
  
  
• "When we are surprised by a particular outcome or event, we should consciously acknowledge that there must be a gap between our perception and reality. ... we should always reexamine a surprising situation from various angles and points of view until that surprising feeling is replaced by a rock-solid intuitive understanding..." [p. 179]
  
• A 3-foot long string of DNA fits into the nucleus of every cell. It does this by a process similar to the twisting of a telephone chord called supercoiling. When the DNA is involved in reproduction, it is too tightly coiled to simply split down the center and recombine with a second molecule. The actual process is much more complex involving some cutting and combining of each of the two strands. [p. 183 - 184]
  
  
• "Another important life lesson that mathematical thinking offers us in our everyday lives is a process for generating new ideas and becoming more creative. That procedure simply entails taking a notion and considering variations on a theme." [p. 191]