Geometry and the Imagination Copyleft* 2009
----------------------------
Math 462 at the University of Massachusetts, Amherst (Spring 2009)
MW 1-2:15 in Lederle 1033 (with a few exceptions)
Prof. Rob Kusner
NOTE: Math 461 is NOT a prerequisite! (However, Math 233, 235 and 300
are recommended.)
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Office hours this semester are after class (till 3ish) and by appointment
- please contact me by email (kusner@math.umass.edu) to let me know if
you plan to come to my office hours, or if you wish to make an appointment
for another time. My office is LGRT 1435G, but often you may find me one
floor above in the GANG Lab (1535 LGRT).
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Bicycle chain magic? Negatively curved vegetables?! Polydrons!?!
What on earth is "Geometry and the Imagination" all about?!?!
Learn more by reading below or by emailing me (kusner@math.umass.edu)
or by signing up and coming to very first class!!!!!
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This is an experimental course in which undergraduates will learn in a
real laboratory setting how to explore basic and profound ideas about
low-dimensional geometry and topology. It is an updated version of a
popular course we tried here four years ago (and a decade earlier),
based on similarly-named courses taught at Princeton, Dartmouth,
Minnesota and elsewhere. The name borrows - indeed steals! - from the
title of a classic book by David Hilbert and Stefan Cohn-Vossen that
described the state of modern geometry in the 1930s. In contrast to
their wonderfully readable book, however, this course is hands-on
(active-learning, discovery-based), so not only will students learn
some interesting mathematics, they also can learn about *how* to learn
mathematics! There is no textbook - students will write their own....
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The instructor is indebted to all those geometers who have preceded
him in this effort; Peter Doyle and John Sullivan in particular gave
the instructor his first hands-on experience in how this hands-on
course might be taught. And since the course is completely hands-on,
CLASS ATTENDANCE - and full participation - IS MANDATORY! To repeat:
this is not a course for the absentee student; YOU - and your open
mind - NEED TO BE PRESENT for the course to be successful - for you,
and for everyone!!
Students - and the instructor - will often need to procure the right
materials for some of the projects to work. Among the most important
things to get (immediately, please) are a lab notebook (bound, not
looseleaf or spiral) and some colored pens. The details of the
notebook are not so important, but it should be big enough for me to
read comfortably on a regular basis, perhaps 100 pages (200 sides),
and not too big to feel uncomfortable packing around with you
everywhere to write ideas down in. (If that's not enough space for
the whole semester, get another when it becomes obvious....)
Bring your lab notebook and pens to class all the time. We will work
together in various groups, as well as individually, so the notes you
take - your rough ideas and false starts, as well as your new (and
perhaps great) insights - will form the basis for evaluation. You,
the student, will be expected to reflect, work out, and expand upon
the ideas germinated together in class, develop these ideas into
coherent, mathematically precise thoughts, and rigorous arguments,
during the rest of the week, and record this ALL in that very same
notebook.
Think of this as a lab course, except it's for a different kind of
experimental science - mathematics - where the role of experiment is
played by proofs, including the formulation of definitions, lemmas and
illustrative examples. Indeed, just as experiment is the ultimate
test for reality in other sciences, so mathematical reality is
ultimately tested by working out proofs.
We'll do this as rigorously as possible, recognizing that ultimate
rigor, in something as rich as mathematics, is itself logically
problematic (thanks to Kurt G\"odel and his successors). So our
emphasis will be to approach rigor "from the top down" (like birds and
real mathematicians) rather than "from the bottom up" (like worms and
pedagogues).
At this point, you may have reasons to be fainthearted - DON'T BE!
We'll have fun in this class, and whatever your future plans - doing
more math, teaching, doing other sciences or writing or carpentry or
whatever - there'll be "relevant" stuff to take away with you. Just
be prepared to work hard - and "smart" - to expand your mind....
And as far as evaluation: I already mentioned that I look forward to
perusing your notebooks from time to time. But also I think it will
be fun and instructive for you all to do a final project for the course
that will involve a presentation to the others in the class, as well as
something written for me to keep. So as the semester progresses, you
might start thinking about a possible topic, and we should certainly settle
on one by Spring Break.
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Mon 26 Jan
How do we use a cloverleaf intersection to make a left turn? If the
usual right turn ramps are missing (or if we forgot to use them), how
do we make a right turn? (This illustrates the old adage: two wrongs
don't make right - but three lefts do!)
We made the convention that left-turning is positive and right-turning is
negative (as in politics, at least half the time :) and found that going
all the way around the cloverleaf amounts to -3 turns. Work out the total
turning for all the possible turns in a cloverleaf.
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Wed 28 Jan
SNOW DAY!!!
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Mon 2 Feb
We explored the turning number for closed and open curves, using the
circle (or "compass") of directions indicated by the unit tangent vector
to the curve.
We related the classical fact that the sum of the interior angles of
a triangle is 180 degrees (1/2 turn) to turning number, and used it
to generalize for N-gons (you guys gave some nice alternative arguments
too)! What happens if the N-gon isn't simple (has edges which cross)?
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Wed 4 Feb
We argued a couple ways that the turning number of a figure "8" curve
must be zero. We also described turning number using calculus ideas
as total curvature (integral of curvature with respect to arclength).
You should explore this in more detail using formulas. Find an
example with infinite turning angle but finite length.
We saw that for closed curves, turning number can be computed by
looking at how many times the tangent vector points east (or any
fixed direction) and counting these with a plus or minus sign depending
on whether the curve is smiling (bending left) or frowning (bending
right) at such points.
A conjecture is that simple closed curves must have turning number 1 or -1
only - you guys are working on a proof of this for homework!
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Mon 9 Feb
We consider some open curves (spirals) with infinite turning number - some
had infinite lenth and some (e.g. the self-similar ones) had finite length.
For spirals of the form r(t)*(cos 2\pi t, sin 2\pi t), t > 1, what condition
on the integral of r(t) from 1 to infinity will guarantee finite length?
We also found a (self-similar) closed curve (continuous image of a circle)
with infinite turning number and finite length.
Using real bicycle chains we experimnetally explore which closed plane curves
can or cannot be deformed to simple closed curves. The handout (courtesy
of John Sullivan) indicated some examples to try: when the turning
number was NOT 1 or -1, what did we discover?
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Wed 11 Feb
After seeing part of the _Outside In_ video, we continue exploring the
deformations of immersed curves using real (and "thought") bike-chains.
Keep in mind the issue of "bike-chain magic" - that is, because a
chain is not really a plane curve, there is either an over- or
under-crossing at each place where the corresponding plane curve would
cross. Without "magic", what rules for over- and under-crossing also
need to be satisfied to deform the chain to standard position?
Finally, there is the issue of knotting: the example on the handout in
the lower right of Figure 2 is an overhand (or trefoil) knot: the
corresponding plane curve has turning number 2 (or -2); you can add a
kink to make this 1 (or -1); with "magic", the resulting curve can be
deformed to standard position (draw a sequence of figures - like a
flip book - to illustrate this deformation); but without "magic" the
knot persists....
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Mon 16 Feb
[honor Washington and Lincoln and...]
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Wed 18 Feb [emergency meeting in 1535=GANG]
We explored how to assign a handedness to each crossing in analogy to
a screw-thread. The writhe of the plane curve (with crossings) is the
number of right-handed crossings minus the number of left-handed
crossings. In other words, assign a "+" for each right-handed
crossing and a "-" for each left-handed crossing and add them up (with
signs). We discovered that these signs didn't depend on how the curve
was rotated in 3-space, nor on the orientation of the curve (i.e. on
which direction we trace the curve), and thus the same is true of the
writhe. But what happens to the writhe if we take the mirror image curve
in 3-space (i.e. if we reverse each of the crossings)?
Conjecture: two bike chains (plane curves with crossings) can be
deformed into each other without using "bike chain magic" if and only
if they have the same turning number, writhe and knot type.
We introduced the notion of the oriented connected sum C#D of closed
curves C and D in R^2, and for homework you're going to figure out
a relationship between their turning numbers T_C, T_D and T_D#D.
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Mon 23 Feb
We discussed oriented connected sum C#D of closed curves C and D in
R^2, using a band whose boundary has turning number 1. We discovered
that the turning number T_C#D = T_C + T_D - 1; in other words (T-1) is
additive under connected sum. Please verify these for as many
examples as you can, and see if you can make a proof (one suggestion
from the class was to think of this operation as introducing a pair of
U-turns).
What happens to this above turning number formula if we use the
"oppositely oriented" connected sum - that is, if we join C and D
using a band whose boundary has turning number -1?
How does writhe behave under #? If a curve bounds a surface in the plane,
what is its writhe...?
Try using the idea of connected sum - or rather something like its
inverse operation, where the curve is cut apart, to show that simple
closed curves must have turning number 1 or -1. Another way to do
this was suggested in class: make a fine grid in the plane, then deform
the curve a little to lie on the grid, arguing that the turning number
doesn't change under the deformation; then deform one square at a time
till all that is left is a single square....
We reviewed the Whitney-Graustein theorem: closed curves C and D in
R^2 can be defomed (by regular homotopy) into each other if and only
if T_C = T_D. Again, try to verify this with examples, or even see if
you can prove the "only if" direction (the "if" direction is hard, and
one proof due to Thurston was the subject one of the OutsideIn video).
A consequence of the Whitney-Graustein theorem: a circle cannot be
turned inside out (T=1 is not the same as T=-1)! We illustrated with
bike chain why a naive attempt to do this leads to a pair of (illegal
for regular homotopy) infinitely tight kinks. As we saw in the
OutsideIn video, the analogous "eversion" can be done 1 dimension
higher: a 2-sphere in R^3 can be turned inside out!
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Wed 25 Feb [emergency meeting in GANG=1535]
[Please try to get some POLYDRON pieces of your own: "A2Z" in Northampton
may have some - call first - or you might find them on the web by googling
"polydron"]
We experimented with POLYDRONS to find turning numbers for closed curves
on polyhedra. The turning number for a simple curve which surrounds a vertex
differs from 1 by the "gap" angle at the vertex - what we see by flattening
the polyhedron into the plane (here we orient curves to go counterclockwise
around the vertex to avoid sign issues):
Turn = 1 - Gap
Please note: if the there is an overlap instead of a gap, we consider Gap
to be negative!
Using POLYDRONS, how close to 1 can you get for Turn without being
exactly 1? (How small a Gap can you find - positive or negative?)
Given the regular polygon pieces in POLYDRONS, please check that this
amounts to determining small values of 60 - 18a - 15b - 10c, where
a, b, c are the numbers of pentagons, squares, triangles, respectively,
surrounding the vertex (a hexagon counts as 2 triangles - why?)!
Does the turning number change if the curve is deformed through curves
that don't pass through any vertices?
What happens when the closed curve surrounds more than one vertex?
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Mon 2 Mar
[Another SNOW DAY!!]
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Wed 4 Mar
We continued experimenting with turning number of curves and gaps at
vertices of polyhedra using POLYDRONS. In particular we found values
of a, b, c which led to gaps of 1/30 and -1/60, and convinced ourselves
that this is the best possible. (Note that although 10, 15 and 18 are
relatively prime, the contraint that a, b, c are non-negative integers
means we can't necessarily find a combination which gives 1/60 - but it
also restricts the choices to a finite set of a, b, c that need to be
considered.) John found an Archimedian polyhedron (using a dodecagon,
a piece not available in POLYDRONS) with a 1/60 gap. I also poined out
that a dihedron made from a pair of regular n-gons has vertex gap 2/n,
which can be made as small as one likes; I also observed that if we allow
any shape polygons (like skinny triagles) we can make closed (irregular)
polyhedra with all gaps arbitrarily small.
Notice that if we rewrite the equation above as:
Gap = 1 - Turn
this should remind you of the quantity (up to a sign) that was additive
under connected sums. This quantity - the Gap - clearly adds: when a curve
surrounds more than a single vertex, the total Gap along the curve is the
sum of the Gaps at each vertex. (Convince yourself of this with POLYDRONS!)
We call this total Gap the Curvature of the region enclosed by the curve:
The Curvature of regions is additive, and
for polyhedra, equals the sum of the Gaps
at the vertices contained in the region!
Careful: if the curve bounding region passes through a vertex, we have a
problem defining the Curvature of the region - we'll assume for now that
this doesn't happen :-)
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Mon 9 Mar
We began by discussing possible project topics - please take a look
at the sheet or use your own imagination to come up with some tentative
topics which we can begin to discuss next class individually - the
spring break would be be a good time to get started on this and to decide
whether each project "fits" each student.
Then we went back to discussing total curvature of a region R on a
polyhedral surface, and we reformulated the above as:
TotalCurvature(R) = \Sum_{v \in R) Gap (v)
and observed that as the boundary curve \dR (oriented, as usual, so that
R lies to the left) is deformed to pass across a vertex v, the turning
number of \dR changes by -Gap(v), proving:
TotalCurvature(R) = 1 - Turn(\dR)
at least for regions R which are disk-like.
We made some polyhedra that close up, like a tetrahedron, or cube or
whatever: what is the Total Curvature for your examples? We found in
all examples so far that the total Curvature of a sphere-like
polyhedron is 2. This jibes with the above formula since if R is a
disk-like region R which covers all vertices of your polyhedron and
whose complement is a little disk on a face, then we checked that
Turn(\dR) = -1 since \dR must go clockwise in order to keep R on its
left! (This proof was discovered by students in my 2005 class :-)
What happens if your polyhedron is like an inner-tube (a torus)?
What if is like the surface of a pretzel withseveral holes?
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Wed 11 Mar
Curvature of vegetables (and fruits):
We had some fun with oranges and kale in class. In fact, we found the
total curvature of the whole orange was (pretty close to) 2. Why do
you suppose that an orange (or cabbage) is positively curved and kale
is negatively curved? One idea has to do with growth rate of cells
along the edge of the leaf.
Here's some stuff to contemplate over the break:
It turns out that if a surface has constant curvature K, then the
circumference C(r) of a circle of radius r grows like
c(K) sin \sqrt{K} r for K > 0
2\pi r for K = 0
c(K) sinh \sqrt{-K} for K < 0
where c(K) = 2\pi/\sqrt{|K|}.
Use L'Hospital's rule to check that as K approaches 0 both expressions
give the familiar formula for the euclidean plane.
Also check that the formula is correct for K > 0 - this jibes with the
geometry of a sphere of radius R, since the curvature density of such
a sphere is exactly K = 1/R^2, since the total curvature is K times
Area = 4\pi radians = 2 turns!
The "hyperbolic" case (K < 0) needs more theory than we can develop in
this course to prove, but it makes sense that C(r) would grow like an
exponential function of r (recall sinh t grows like e^t), since around
each point of space, negative curvature ("negative gap") means excess
surface....
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Mon 23 Mar
Curvature of cabbage (is that a fruit, or a vegetable - or a former
German Chancellor?! ;-):
What did you get for the curvature of your various cabbage leaves?
Were you able to "develop" strips along curves on a sphere (the red
or yellow 4-square, 3-ball, 2-sphere) to their corresponding strips
in the plane? Did you make any observations about how the total
curvature is related to the area (or area fraction) of the region on
the sphere bounded by your curve?
(Laura, Liz and Katie discussed details of their potential projects with
Rob as well. Hyperbolic sleeves and trousers; Platonic kaleidoscopes or
panoramas; and delta-square-hedra, respectively, were proposed so far.)
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Wed 25 Mar
Spherical geometry:
We figured out that the area fraction a of a spherical triangle with
interior angles x, y, z are related by
x + y + z = 1/2 + 2a
which agrees with the euclidean limit (as a tends to 0).
Show that if instead we use turning angles, the total turning number
satisfies t = 1 - 2a, where we surround an area to the left of the
triangular curve. This makes sense for small triangles (a nearly 0).
Check that this works not only for triangles, but for any simple
polygon - and by a limiting argument, for any simple curve on the
sphere! In fact, for an equator (or any curve surrounding area
fraction a = 1/2) we see t = 0 as expected.
(Lubna discussed with Rob a potential project about tiling hyperbolic
surfaces using right-angled pentagons.....)
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Mon 30 Mar
We began to explore (in the case of regular polyhedra, like tetrahedron
or cube) why the curvature (= gap) at a vertex equals the sherical
image area (normalized so that the whole sphere has area 2, in units of
turning).
Work out the area of spherical image for any vertex of a regular
polyhedron in two ways: by dividing 2 by the number of vertices, and
using the detailed geometry - does the answer agree with the gap?
Now try this for a general polyhedral vertex, using the fact that
the perpendicular plane to any edge is what cuts out the great circle
arc of the edge's spherical image.
(Rob continued discussion with Lubna on approximating hyperbolic
surfaces with squares meeting five to a vertex, and also discussed
with Chris about moduli of elliptic curves and hyperbolic orbifolds in
general....)
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Wed 1 Apr (no foolin'!)
Today we began to "put it all together" by working out that spherical
image area (extrinsic) is really the same as (intrinsic) total
curvature (as measured by gaps) for polyhedra. This leads, in effect,
to two deep theorems of Gauss rolled into one: Gauss's Theorem
Egregium and the "Gauss-Bonnet" Theorem!
Gauss's Theorem Egregium asserts that the intrinsic and extrinsic
curvature measures agree. For "Gauss Bonnet" we need to recall the
notion of Euler number of a region R on a surface, denoted Euler(R) or
\chi(R). On a polyhedron, by definition
Euler(R) = V - E + F
where V, E and F are the numbers of vertices, edges and faces,
respectively. We explored why this is independent of the subdivision
into polygons, and why it makes sense for surfaces with more general
cellular subdivisions. We also saw that Euler number is additive under
disjoint unions and used that to compute what happens under "connected sums"
and "surgeries" of surfaces (where a pair of disks is cut out and an annulus
is glued in, reducing Euler by 2).
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Mon 6 Apr
And now for the combined Theorem Egregium and Gauss-Bonnet:
As we saw earlier when R is a disk-like region (where Euler(R) = 1)
the formula for the curvature measure of a region on a surface in
terms of turning angle of the boundary gives:
TotalCurv(R) + Turn(\d R) = Euler(R)
where \d R is the boundary of R.
Try to do the general case using the disk case as a model!
This can also be thought of as Gauss's Theorem Egregium (intrinsic
curvature agrees with extrinsic curvature), since (in case R is a
disk) we can compute Turn(\d R) in two ways:
Turn(\d R) = 1 - TotalCurv(R)
as we know from a long time ago (or see last time with Euler(R) = 1).
In the special case of R being the neighborhood of a one vertex on a
polyhedron, this is just
Turn(\d R) = 1 - gap.
On the other hand, in the case of a vertex with three faces meeting
it, whose normal image N(R) is thus a spherical triangle with interior
angles x, y, z (you can try the more general vertex, which invlolves
not a spherical triangle, but spherical polygon as normal image) we have
(by "developing" or rolling on the plane):
Turn(\d R) = Turn(\d N(R)) = (1/2 - x) + (1/2 - y) + (1/2 - z)
= 3/2 - (x + y + z) = 1 + (1/2 - (x + y + z))
= 1 - 2a = 1 - A
where a is the area fraction (or A is the area) of N(R). Thus the intrinsic
curvature equals the extrinsic curvature:
gap = A.
Try to do the same computation using spherical trigonometry directly instead
of this developing/rolling trick.
(Chris discussed his sundial for an ideal general planet project with Rob.)
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Wed 10 Apr
Here's another way to compute the Euler number of a surface in space.
It uses height functions and should remind you how turning number of
plane curves (1D in 2D) could be computed using "smiles" and "frowns"
- this idea was mentioned in the _OutsideIn_ video.
First, note that in addition to 2D-smiles (pits) and 2D-frowns (peaks),
which can be thought of as critical points (local minima and maxima) of
a height function, there is third version - a kind of 2D-frown/smile
(pass) - of critical point (some call this a saddle point because a
neighborhood of the surface resembles a horse's saddle).
To find an analogue of the turning number which is invariant under
deformations, we looked at some examples: if a sphere has an indentation
near its peak, a new pit and pass are created; if a "horn" grows from
the side of a sphere, a new peak and pass are created. And if we rotate
a sphere, what was a peak becomes a pit and vice versa. This suggests
that the number
peaks - passes + pits
should be invarant under deformations.
Compute this for lots of deformed shapes of spheres - do you always get 2?
(What happens if there are infinitely many critical points?)
Now try computing this number for a bagel, a pretzel, etc.
If we keep track of the connections (gradient flow lines) between critical
points, this gives a decomposition of the surface into cells of dimension
0, 1 or 2, analogous to the decomposition of a polyhedron into V, E or F.
And how does all this compare with what we did with total curvature and
Gauss normal spherical images? What is the *sign* of the area of the normal
image near a pit or peak? What about near a pass?!
All this should convince you that
Euler = peaks - passes + pits
for a closed surface. What about when we have a region R with nonempty
boundary \dR? (Try to figure out a formula that corrects for critical
points of height restricted to \dR!)
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Mon 13 Apr
We considered the Mobius classificication of surfaces. If we remove a disk
from a sphere, what's left is a disk. Removing more disks is the same as
attaching (untwisted) bands to a disk. Mobius' idea is to do something
similar for a general surface:
Start with disk, which has k=1 boundary components. Attaching bands to the
boundary circle in an "unlinked" way increases k by one for each band.
Suppose we want to increase the genus g (recall this is the maximum number
of disjoint simple curves which fail to separate the surface) by one. We
can do that by attaching a pair of bands to the boundary whose attaching
regions are "linked" on the circle (that is, if we look at how the the core
curves of the bands intersect, we get an odd number, unlike the case of
unlinked bands above, whose core curves intersect an even number of times).
Finally, we can attach m half-twisted (a.k.a. Mobius) bands to tbe boundary.
Mobius's classification theorem is that every surface can be realized this
way. See if you can decide when (g, k, m) and (g', k', m') yield equivalent
(homeomorphic) surfaces. This involves sliding one band along another.
Back to Morse height functions: We also saw how to associate an h and its
gradient field to a (polyhedral) surface in such a way that pits, passes
and peaks correspond to (midpoints of) vertices, edges and faces - that
confirms our Euler number speculations from last time. We also discussed
how critical points on the boundary might be counted (with 1/2 weighting).
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Tax 15 Apr
We discussed a little more about the Mobius classification of
surfaces, and noted three things: 1) there more information if we
consider the surface to be immersed on the plane, namely, the even or
odd number of full twists in the (ordinary) bands (can you think what
might be done with the Mobius bands along the same lines?); 2) if (g,k,m)
is finite, then when m > 0 there a relation g+1+m=g'+3m' giving another
surface of type (g',k',m') homeomorphic to the first; if this triple is
infinite then g (and m) must be countable, but k may not be....
We gave an alternative proof of Euler number of a sphere = 2 using
"charges" and a puff of west wind (a vector field w on the sphere) in
order to blow all the charges a bit to the east (except for the two
positive charges at the N and S poles. Because the number of edges
(where we placed negative charges and were careful to not align them
on the east-west axis) on the west side of any face is one more that
the number vertices on the west side of the same face (we put positive
charges at the vertices, as well as on each face), the windblown
charges (now all on the interior of the faces, except at the poles, of
course) sum to zero on each face, and thus to 2 althogether. On the
other hand, the total charge just counts V - E + F = Euler!
We also see this works on a torus (Euler = 0), since then we can find a
vector field w which never vanishes.
Check that whether this works (mutatis mutandis) on any closed surface
of any genus.... (Something should puzzle you: can we get Euler < 0?)
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Fri 24 Apr ...... [still being edited]
Observe that Total curvature = Euler # by splitting charges in the
obvious way: the edge charges split into -1/2 for each adjacent face;
the vertex charges split into the fraction of angle associated to each
face, leaving the "gap" or curvature of the vertex as the net charge their.
The total charge in each face is again zero (this time by looking at the
tunring number of the boundary of each face) and the total curvature
is thus the total net charge (which is the Euler # by the above).
Explore a little more about wind patterns on surfaces and
discover that the total number of ZEROES (of the vector field, where
the wind stops, where the "hair" has a "cowlick" ;) also equals the
Euler #(something we alread saw for gradient vector fields.
This number of zeroes is counted with an index that is +1 for a sink or
source and -1 for a saddle - and more generally is the number of times
the field's direction winds around 0 when we follow a little loop around
the place where it vanishes.
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Mon 27 Apr ...... [still being edited]
Orbifolds and their (fractional) Euler numbers
The tile patterns on the hall floor, the brick wall and the Campsu
Center patio, and their orbifolds (17 of them).
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Wed 29 Apr ...... [still being edited]
The 2-orbifolds for various platonic polyhedra.
Unifolds (1-orbifolds - we found 4 connected examples, including R
with trivial symmetry; R/-=ray; R/Z = S^1 and R/D=S^1/-=interval).
In considering the last of these, we need to clear up on confusion:
the glide-reflection group is not cyclic (it's not even abelian - it's
actually the infinite dihedral group) and so it is not generated by a
single element (as I was trying to make it be); the two generators
are reflections S(x)=-x and R(x)=1-x, and the their composition is the
translation T(x)=R(S(x))=1+x. (I'm not quite sure what the hop-twists
that I did across the stage were, but maybe you can figure that out? :-)
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Mon 04 May and beyond.... [still being edited]
We defined the orbifold Euler number using charges, and we explored
the teardrop T_n and cigar C_n = T_n # T_n, which have orbifold Euler
number 1 + 1/n and 2/n, respectively. What other orbifolds with local
structure R^2/Rot have positive orbifold Euler number? (Hint: first
decide how many cone points are possible!?!)
We explored the orbifolds T_a # T_b # T_c associated with the
(rotation) symmetries of the various platonic solids, including the
infinite family of "dihedra" (plural of "dihedron", where [a,b,c] =
[2,2,n]), the tetrahedron [2,3,3], the cube (or its dual, the
octahedron [2,3,4]) and the dodecahedron (or its dual, the icosahedron
[2,3,5]): these give rise to the dihedral groups D_n, the tetrahedral
group A_4 (even permutations of 4 objects), the octahedral group S_4
(all permutations of 4 objects) and icosahedral group A_5 (even
permutations of 5 objects). These orbifolds, together with the
teardrops T_n, their connected sums (including the cigars C_n = T_n #
T_n, which correspond to cyclic group of symmetries, also denoted C_n)
are the only ones of this form with positive curvature, i.e. positive
(orbifold) Euler number. All the rest give rise to infinite symmetry
groups, such as the frieze or wallpaper groups (the orbifolds are
flat, i.e. have Euler number 0, such as [2,2,\infinity], [2,3,6],
[2,4,4], [3,3,3] and [2,2,2,2]), or the hyperbolic (Klein) groups
corresponding to orbifolds like [2,3,7] or even the "ideal
homentaschen" [\infinity, \infinity, \infinity].
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All material copyleft by Rob Kusner, 2005-2009
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Project Presentation Schedule for Spring 2009
6 May @ 1
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Katie M: Deltahedra
Lubna M: Tiling Hyperbolic Surfaces
11 May @ 1
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Chris E: Geometry of Sundials
Andrew R: Plane curves, Arnol'd invariants and Blank words
13 May @ 1 [meet at 1535]
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Liz K: Panoramic polyhedra
Christina B: P#P#P = T#P (and other amazing surface topology)
Ian C: Schwarzian derivatives
Spencer R: ?????
18 May @ 10:30 [meet at 1535]
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James: All the way with Gauss-Bonnet (well, almost)!
Laura W: Hyperbolic crocheting: strips, cusps and pseoudosphere
Chris B: Hypercube model - see photo
Jess R: Klein surface realizing the Hurwitz 84(g-1) bound
*"Copyleft" means that you have permission to use this material for
non-commercial purposes as long as you acknowledge its source. Any
use, in whole or in part, must include the "Copyleft 2009 by Rob
Kusner" message or its equivalent. Commercial use without prior
approval of the copyleft holder is strictly forbidden, and is
punishable by methods not even imagined by John Yoo!