An Example of a "Learning Process" Journal (using the 2 colored box format)

"The Road to Reality " by Roger Penrose (Chap. 2 An ancient theorem and a modern question. p. 25 - 50)

Quotations:

• "An important message of the discussion in the preceding sections is that the Pythagorean theorem seems to depend on the parallel postulate. ... Let us try to address the question of what would happen if the parallel postulate is indeed allowed to be taken false." [p. 33]
• "It [Escher's woodcut called Circle Limit I] actually provides us with a very accurate representation of a kind of geometry - called hyperbolic (or sometimes Lobachevskian) geometry - in which the parallel postulate is false, the Pythagorean theorem fails to hold, and the angles of a triangle do not add to p." [p. 33]
• "However, there is actually something particularly elegant and remarkable about what does happen when we add up the angles of a hyperbolic triangle: the shortfall is always proportional to the area of the triangle. ... In Euclidean geometry, there is no way to express the area of a triangle simply in terms of its angles." [p. 35]
• "What, then, is the observational status of the large-scale spatial geometry of the universe? It is only fair to say that we do not yet know ... the issue is still fraught with controversy ... " [p. 48]
• "Fortunately for those, such as myself, who are attracted to the beauties of hyperbolic geometry, and also to the magnificence of modern physics, there is another role for this superb geometry that is undisputedly fundamental to our modern understanding of the physical universe. For the space of velocities, according to modern relativity theory, is certainly three-dimensional hyperbolic geometry." [p. 48]

Questions:

• What is the mathematical idea expressed by the term conformal? Penrose seems to say that it is any representation that preserves the idea of angle is a conformal representation. This corresponds to the definition I found by googling 'conformal mathematics'.
• Are there introductory online sources about hyperbolic geometry?

References:

Web Sites:

• http://mathworld.wolfram.com/ConformalMapping.html
  
• http://www.math.clemson.edu/~rsimms/triangle/hyperbolic.html
  
• http://www.geom.uiuc.edu/docs/forum/hype/model.html

It has been 4 days since I began making notes for this book. In that time I have read and yellow highlighted chapter 2 and have had a quick skim read of chapter 3. I have also been diverted into a beginning of symmetry by reading parts of the first few pages of "Group Theory and Chemistry", "Symmetry in Chaos" and Handbook of Regular Patterns".

This morning (7 am) I have decided to return to Penrose and continue making notes. One of my first ideas is to incorporate the idea of a Key Idea into the note box (green) by inserting a new box for this and giving it a new color. From there is is a simple idea to also include an icon that acts as a link to an Inspiration concept map for the chapter. This took a little while as I had to upload copies of two software programs ImageForge and SnagIt. But I now have all of my regular software installed on my laptop.

I have then added 4 sub-headings: Quotations, Questions, References and Web Sites. Now to add content to each of these headings as I reread the chapter and note the yellow highlights.

I love Penrose's writing: for example, "there is actually something particularly elegant and remarkable about what does happen when we add up the angles of a hyperbolic triangle". He captures the aesthetic and affective aspect of mathematics, which is critically important. Mathematics is not bloodless. Perhaps the Platonic mathematical universe "exists" and is simply a logical system, but the exploration of that system with our discoveries of relationships and patterns is intrinsically human. It is this process that is the joy, not the results. This is where we are miseducating our youth. We are trying to teach them to be little robots, blindly following algorithmic routines instead of teaching them to explore and play.

Googling 'hyperbolic geometry introduction' gave a treasure trove of hits.

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