Course:Harris, Fall 08: Diary Week 15

Mon:

• I mentioned some issues on the exam corrections:
  • The idea is to show mathematical details as to why some regular tessellations are possible and others impossible (focussing just on what happens at a vertex).
  • The point is to show that you understand the mathematical details, how it works out exactly, why it works as claimed.
  • Resubmissions will be allowed, both for those who really hadn't a clue on current submission and for those who wish to improve based on specific comments I made on their submissions.
• I mentioned that the Final Exam would likely include an expansion of #6 from Exam 2: Show that there are exactly 5 regular tessellations of the sphere.
  • Basic idea is to look at how you can make a regular tessellation by fitting $k$ polygons around a vertex.
    • That tells you that the angles of the regular polygon are each 360/$k$ degrees.
      • From that you can calculate what the angle-sum would be, if you were doing a regular tessellation with
        • triangles
        • quadrilaterals
        • $n$-gons.
    • On the other hand, you know something about what angle-sum has to be for
      • a spherical triangle
      • a spherical quadrilateral
      • a spherical $n$-gon.
    • So just compare those two to show what works and what doesn't work.
• I mentioned the required paper on "What is the shape of space?"
  • This is to be thought of as a philosophical piece.
  • Consider the shape (i.e., geometry) of astronomical or cosmological space, not human-scale space.
  • If you draw on outside sources for ideas, be sure to credit them; but you should be able to approach the question with just the material we've covered in class.
  • One aspect we've not talked about a lot is how to think about 3-dimensional space, since our approach has been to study 2-dimensional geometries.
    • Some things don't change when going up to three dimensions:
      • Angle-sums for triangles obey the same properties--Euclidean, spherical, or hyperbolic--irrespective of 2- or 3-dimensional space, since any triangle lives in a 2-dimensional slice, anyway.
      • Similarly, if you consider the way parallel lines in a 2-dimensional slice behave, you'll get the same Euclidean, spherical, or hyperbolic behavior within that slice.
      • Just as in 2 dimensions, a 3-dimensional spherical space is finite, whereas 3-dimensional Euclidean and hyperbolic spaces are infinite.
        • The earliest exposition of a 3-dimensional spherical space is from Dante's Paradiso, in which he conceives of the universe as
          • having a center at the center of the Earth (where the devil is trapped in ice),
          • surrounded by increasingly larger 2-dimensional spheres of Hell,
          • surrounded by the 2-dimensional sphere of the Earth's surface,
          • surrounded by large 2-dimensional celestial spheres
          • followed by various angelic spheres, with the higher-order ones decreasing in size (fewer seraphim than cherubim, etc.)
          • culminating in the Godhead seated at a single point, the center about which all the angelic spheres are concentric.
        • This is comparable to considering a sphere as being built up from the South Pole, surrounded by small circles of latitude, then larger ones, then the equator, then shrinking circles of latitude, then the single point of the North Pole.
• Groups spent 20 minutes doing the Dilation Exploration and starting the Iterations Exploration.

Wed:

• We talked about what sort of knowledge we have of the shape of the universe.
  • A question was raised about why the universe is often assumed by physicists to be isotropic.
    • Isotropic means the same going out in all directions.
    • This is related to the assumption that the universe is homogenous (same at all points).
      • The reason for assuming homogeneity is that we have no reason to believe we're in a special point of the universe--whatever we see here is probably the same as everywhere else, more or less.
    • Since from where we sit, the universe appears pretty much the same in all directions, we figure the universe is generally isotropic.
    • The reason for worrying about homogeneity and isotropy is that the only geometries which are homogeneous and isotropic are the three we've studied:
      • Euclidean (or flat or zero curvature)
      • spherical (or uniform positive curvature)
      • hyperbolic (or uniform negative curvature)
  • A question was raised about how we can know, with any precision, what the universe is like.
    • On the near scale (say, the nearest dozen galaxies or so), we have pretty good measurements for distances, velocities, and masses of stars and galaxies (at least, the visible stars within galaxies).
      • In particular, we can compare how fast stars move within their galaxies with the gravitational forces they're subject to from the visible stars.
        • Those don't add up: There must be a lot more mass in each galaxy than just appears among the visible stars (the stellar mass).
        • This has been known since the 80s or so; the non-stellar matter is called Dark Matter (since it's not visible).
        • The Dark Matter is actually by far the greatest component of galaxies, totaling some five or ten times the stellar matter.
        • Most surprisingly of all, recent theories and observations seem to rule out any kind of ordinary matter for the Dark Matter: not dust, not burned-out stars, not any of the usual structures we know of.
        • Best guesses currently are that the Dark Matter is made up of "strange" sub-atomic particles, not the ones forming ordinary atoms. Nobody knows why it should be here in such overwhelming proportion.
    • On the far scale, we rely mostly on numbers of galaxies, their distances, and their velocities with respect to us.
      • Edwin Hubble, back in the 20s, discovered that distant galaxies have light which is red-shifted.
        • That means that the characteristic signatures of various elements show up as having longer (redder) wavelengths than they do coming from, say, the sun.
        • The only explanation for this is the Doppler effect:
          • Just as a siren on a police car speeding away from us sounds at a lower pitch (longer wavelength), the light we see from a star speeding away from us has lower wavelengths than it would otherwise.
        • That means that the distant galaxies are all receding from us.
        • Even more interesting: The farther the galaxy, the quicker it's receding from us.
        • This is called the expansion of the universe: Every point of the universe is rushing away from every other point, much like a balloon that is expanding.
      • The standard cosmological models (assuming homogeneity and isotropy) accommodate this expansion:
        • If the universe is positively curved (i.e., spherical), then it starts from a Big Bang, expands, decelerates, comes to a maximum size, then starts to shrink, eventually resulting in a Big Crunch.
        • If the universe is negatively curved (i.e., hyperbolic), then it starts from a Big Bang, expands, decelerates, but keeps expanding at a slower and slower pace, never stopping.
        • If the universe is flat, then it starts from Big Bang, expands, decelerates, and keeps expanding at a slower and slower pace, much like the hyperbolic case (only slower).
      • These models all have a Big Bang, because we can see the remnant of that in the Cosmic Background Microwave Radiation, which we've known about since the 60s.
      • What is new--just discovered in the last decade or so--is that the universe's expansion is accelerating.
        • This is very surprising (and contrary to the standard models above), as one would expect gravity to exert a contracting force, slowing down the expansion. Apparently, something else is counteracting the contracting force of gravity.
      • Nobody has any explanation for this. It's often called Dark Energy, but that's more of a play on Dark Matter than any kind of explanation.
        • Einstein's equations allow for this kind behavior, by means of an uknown constant, called the Cosmological Constant.
          • Until recently, it was believed the Cosmological Constant was probably 0, as no one could imagine any reason it shouldn't be 0, and observations were consistent with that.
          • If the Cosmological Constant is 0, then the universe's expansion has to be decelerating, and that's what was assumed until recently.
          • It is only the more recent observations that contradict that, suggesting the Cosmological Constant isn't 0, after all; but no one knows why that should be.
  • Cosmology is not a robust science; it experiences radical changes in its operating paradigms, roughly every couple decades.
    • We live in very interesting times!
• We looked at fractals.
  • Fractals are objects which retain the same general appearance upon expansion.
  • Examples from nature:
    • coastlines
    • cloud surfaces
    • leaf edges
    • water waves
  • It is the fractal nature of much of the physical and biological world that makes it possible to render images with digitally created landscapes, oceanscapes, and so on, as such images can be created with just a little bit of programming.
  • Fractals have a relationship with dimension in the following manner:
    • Consider taking a figure and dividing it up into sub-figures, by reducing the side-lengths by (say) 2:
      • dimension 1:
        • A line segment is divided so that the sub-segments have length 1/2 the original.
          • We end up with 2 sub-segments.
        • Repeating on the sub-segments: Divide to have smaller segments of length 1/2 as the previous.
          • We get 4 total smaller segments.
          • That's 2 times the previous result.
        • Each subsequent division to length 1/2 the previous results in 2 times as many objects.
      • dimension 2:
        • A square is divided up so that the sub-squares have side-length 1/2 the original.
          • We end up with 4 sub-squares.
        • Repeating on the sub-squares: Divide to have smaller squares of side-length 1/2 the previous.
          • We get 16 total sub-squares.
          • That's 4 times the previous result.
        • Each subsequent division to side-length 1/2 the previous results in 4 times as many objects.
      • dimension 3:
        • A cube is divided up so that the sub-cubes have side-length 1/2 the original.
          • We end up with 8 sub-cubes.
        • Repeating on the sub-cubes: Divide to have smaller cubes of side-length 1/2 the previous.
          • We get 64 total sub-cubes.
          • That's 8 times the previous result.
        • Each subsequent division to side-length 1/2 the previous results in 8 times as many objects.
      • Generalizing to dimension $n$:
        • Each division to side-length 1/2 the previous results in $2^{n}$ times as many objects.
    • Dimension of the object is thus encoded into the result of division by a given factor of side-length.
      • If we find that we expand the number of sub-objects by $N$ when dividing so as to have a ratio of $r$ in side-length, then
        • dimension = log($N$)/log($r$)
          • (In the results above, $N=2^{n}$ and $r$ = 2.)
    • What's that all got to do with fractals?
      • A fractal object, when expanded by some factor $r$ (in side-length), shows the same structure it had before the expansion, but now with some $N$ times as many sub-objects showing. The formula above gives the dimension of the fractal object.
• Groups spent 20 minutes working on the Iteration and Fractal Explorations.

Fri:

• I reviewed the intended meaning of "shape of space":
  • I don't mean a shape, such a sphere, sitting in some exterior environment.
  • Rather, I mean the geometry that the universe obeys, i.e., the geometrical rules obeyed by space, lines, distances, and angles.
  • Examples of the types of rules:
    • What are permissible angle sums for triangles?
    • Are lines infinite or finite in length?
    • In a 2-dimensional slice:
      • Do lines have to intersect, or can there be parallel lines?
      • If parallel lines exist, how many parallels can a given line have through a given point?
      • What kind of regular tessellations are there?
• I reviewed what's meant by a similarity transformation:
  • That is a transformation which can expand or decrease by a scaling factor.
  • We looked at how to measure such scaling factors in a design.
    • Typically, we can say something like "Applying the transformation, something of 1 unit in length shrinks to something .7 unit long."
• We looked more closely at fractal dimension:
  • We recalled that a design is called fractal if it has a similarity transformation:
    • i.e., expanding (or shrinking) it by some scale factor--and perhaps including a rotation or translation or reflection--preserves the same design structure.
  • In the Fractal Exploration, the Snowflake Curve, as exhibited, is wrong, in that the first line should as long as the others.
    • The scaling factor is found by comparing the length of the new segments to the original (properly lengthened!) segment.
    • The multiplying factor is the number of new segments that the old segment is replaced by.
• I mentioned the type of thing that may be found on the Final Exam (which is noon, Friday).
  • That includes just about all of the above (but not including using logs to calculate fractal dimension):
    • differences among the three geometries
    • understanding what fractal means
    • being able to identify similarity transformations, including the scaling ratio
• The groups had 20-25 minutes to finish up the Fractal Exploration and start looking at one more Exploration.
  • We'll finish up on Monday.
  • Shape of Space paper is due on Monday.