Hello, I'm Harel Barzilai, and I guess I'm supposed to start off by summarizing my entire life for you in a few minutes, or at least my mathematical life, which is a bit easier to do.
Well, first off, I'm a post-doc with ITCEP, the center which runs Project Gamma, UMTYMP, and a host of other probjects, and which was formerly known as Special Projects.
I got my PhD in mathematics this past summer at Cornell University, in the field of mathematics known as topology, and in an area which is related to the kind of mathematics we'll be working with today.
I also spent two years during 5th and 6th grades at an "open school" which encouraged each student to explore and discover, and promoted a curiosity-based learning rather than a handed-down-knowlege type of learning, and those early experiences, I believe, are in large part responsible for my success in mathematics, so I'm especially happy to be here tonight to try to encourage mathematical explorations in a discovery, hands-on setting through project GAMMA.
The title, "Mr. Square's Universe and Ours" refers to the famous chacacter Mr. Square in Edwin Abbott's Flatland, because I want to talk about what a universe can look like, but I want to start by having us think about what a 2-dimensional universe might look like, so that we can -- later on -- think about the more difficult question of what our own universe might look like.
Ok, so let's start by trying to think like a 2-dimensional creature -- one having no thickness -- the way they would think about their universe.
First of all, if they see no "end" or "boundary" to their universe, what would be the most likely (and probably the most "natural") shape which they would guess applies to their universe?
That the flatland universe is a plane was the prevailing theory in Flatsville for many years, until one day a Radical Opposing Theory was proposed by a fringe group of scientists. They proposed an alternative shape for the 2-dimensional universe which they called a "sphere" While flatlanders can'treally draw or fully picture a sphere, the sphere proposers imagined it as follows:
and
OVERHEAD #2'
HAND OUT BALLOONS
Question: If, nontheless, you find yourself ending up where you started, what other issue would come up? Answer: Are you sure you really went in a straight line rather than a very, very large circle (which lies in a plane) by means of a nearly but not quite straight path?
Question: What ability(ies) would guarantee that this issue would not arise? Answer: Technological (?) ability to travel in perfectly straight lines.
Question:If you solve these issues AND return to the same spot, after traveling straight, what can you conclude? Answer: The universe is not a plane
Workshop facilitatorto all(?):"[Summarize these observations, then:] Does is follow in that case that Mr. Square's universe must be a sphere?
only after a little bit of this, put up:
HAND OUT INNER TUBES
Pre-Exercise#2 points to make to students: : #1 How do you know that two shapes which look different are really different? Question #2: How do you know you have exhausted all possible surfaces? Question #3 How would a Flatlander (try to) tell which of these was the universe in which they were living?
[Possible Answers: #1 Can pull a string (starting and ending at the same point) tight up to a point and then it cannot be "pulled together" any further. Answer #2: Two intersecting circles on a sphere must intersect in a pair of points, but on a torus you can find two circles which intersect in only one point Answer #3: On a sphere a circle separates: you can't get from one side to the other without crossing/going-through/cutting-open the string, while on a torus you can.
[Facilitator summarizes this...also they would have come up with the existence of genus-g tori by now. Then asks them to generalize these answers to showing that two tori of different genuses are indeed topologically distinct.
OVERHEAD #5: N-HANDLED TORI
OPEN ENDED/TEASER:
Think about at home how you (or flatlander) might tell the difference between two tori of different genuses -- try extending the ideas you used to distinguish the sphere from the 1-torus.
Question: What would a Flatlander astronomer see if they lived
Draw pictures of what you, as a 2-dim'l astronomer in each of these uniserves, would see and if you're inclined, maybe also Diary Entries of this astronomer.
Can you think of what 3-dimensional analogs might be of a sphere? torus? How could you try to create models? How could you mathematically describe them? What would you see as an astronomer if the universe in which we live was the type of a 3 dimensional universe? What other tests might you devise, using rope or other materials, to try to find out the shape of our universe.
Two answers: First, some of these shapes correspond to univserses with "wormholes"
For my second reason as to why we care, I first have a question. Has anyone heard of the Big Bang? [Yes!] Of an "Open Universe versus a "Closed Universe" [briefly review this].
It turns out that there are natural geometries associated to the 2-dimensional shapes ("2-dimensional manifolds", or "surfaces") we've looked at, and there is a notion of "curvature" which comes in under which the curvature of the regular Euclidean plane is ZERO (same for torus), that of a sphere is POSITIVE, and that of a torus with 2 or more holes is NEGATIVE.
There are analogous notions of "curvature" for 3-dimensional "possible shapes for universes" and that whether our universe is open or closed is inextricably linked to the curvature of our universe
Then comment on why it MAKES NO SENSE (not just "a mathematical Mobius band has no thickness" but "we said it was a 2-dimensional universe, not a 3-dimensional, right?"). to talk of "it has only one side" since there are no "sides" really.
What then is unusual about the Mobius band? Use ??? overhead 'paper' and overhead markers, and SCOTCH TAPE. Draw Mr. Square like so:
Sample Diary entries. The impressions of everyone in flatland except Mr. Square and Ms. Triangle, as told by the Daily Flatland Tribune:
[...they left but then came back all RESERVED -- or maybe it is imposters who have come back, or the real square and triangle returned to the wrong universe while reserve-universe triangle and square have mistakenly come to "return" to our universe at the end of their journey. Square and Triangle claim they are the same ones who left here (but how would they know they left the Flatland of THIS universe rather than of another, anyhow?), and that, furthermore they are under the delusion that it is not they but US who have come up reversed!The Diary of Square and Triangle:
When we returned to the exact spot where Flatland was, we found either another flatland or the original flatland RESERVED. All the people there are under the illusion that nothing has changed however, and furthermore they claim that *we* are the ones who have change and come back "backwards"!
[Throwaway trick(?): Show cutting the mobius band down the middle and ask them to try it at home; to try to explain it; and to investigate further cuttings of the band and more-twisted bands and other surfaces.]
Now do the same thing for an analogous trip in a 3-dimensional universe which you and a travel partner take, and further elaborate on the above Newspaper/diary entries by talking about: signs on the road, facial markings, and left- versus right-handedness.