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Or put more formally, this is an elementary construction of the convex regular polytopes of every dimension. This is the presentation I gave at MathFest 2010. For the last part you’ll need some red/blue glasses. These are normally used to create 3d effects, but here we’ll be seeing in 4 dimensions.

I plan on making a video out of this that you may enjoy more.

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A Platonic Solid is a 3 dimensional object that has the following property: select a face, then an edge on that face, then a vertex on that edge. This selection is called a ‘flag’. All the flags look the same and can be sent to each other by symmetries. This definition is really easily generalizable to work in any dimension: for example, if we’re in 4 dimensions, first you select a 3d cell, then a 2d face, then a 1d edge, then a 0d vertex.

The way to construct these objects is to make them in one dimension lower and try to fit them around a vertex (in the case of 3d, an edge if you’re making 4d things, a face if you’re making 5d things, etc) and ‘fold them up’ into the next dimension. To see if you can fit them, look at their interior angle (that’s the angle listed next to each of the polytopes). Any multiple of them from 3 upwards (you need at least 3 so that when you fold them up they don’t just fold flat) that has total angle less than 360 degrees will work. For example, with the triangle, which has interior angle 60 degrees, you can put 3, 4, or 5 of them around a vertex, making the tetrahedron, octahedron, and icosahedron, respectively.

The last section, with the photos, deals with the Universal Regular Polytopes, the ones that exist in every dimension. The Simplex, on the left, is n+1 (n is the dimension you’re in) points that are equidistant from each other. The Cube, in the center, is the Minkowski Sum of the standard basis vectors-that is, you move one of the vectors along another and look at the shape you trace out. The Cross Polytope, on the right, is the convex hull of the positive and negative standard basis vectors. For the full effect on the 4d versions you should wear red/blue glasses. For the 4d simplex you should see a tetrahedron with another point floating in front of it: all of those points are the same distance from each other. For the cube, the ‘big’ cube is not actually bigger than the ‘small’ cube, it’s just closer. For the cross polytope, the blue lines are in a single 3d hyperplane, while the orange lines traverse the 4th dimension: one point is closer to you, one is further away.

I met two of the employees of the Worldwide Center of Mathematics at MathFest this summer. We got to talking and, being as excited and eager to show it off as I was, I showed them my talk. They enjoyed it enough they forwarded it to David Massey, a professor at Northeastern University and the head of the Center. He in turn enjoyed it enough to invite me out to present it at the Center, and on Friday December 17th 2010 I did!

You can watch the recording of the talk on the Worldwide Center of Mathematics‘ website. It’s under the Research tab, in the Worldwide Research Lectures: just scroll down to my name.

Boston is an absolutely amazing city, by the way. I didn’t have nearly enough time there.

Real Analysis I Review

Here’s a presentation I made for my Real Analysis II class. The idea is to refresh your memory about Real Analysis I, go over the important definitions, some theorems, stuff like that. I didn’t have quite enough time to devote to it to really make it complete, but I think it does a fairly decent job.I’m really liking this Prezi site, it makes it really easy to make visually interesting presentations.

Embedding isn’t working, so click this link to see the presentation.

This is the paper version of my MathFest 2010 presentation.

Click Here to Read the PDF

I am going to submit it to various people to read and hopefully edit, and then it will be off to the PME journal to be hopefully reviewed for publication!

This is the paper version of the talk I gave at MathFest 2009. It’s currently being reviewed for publication by the Pi Mu Epsilon journal. Nobody who asked to see it before publication actually responded, so it’s fairly unedited. If you have any suggestions please don’t hesitate to let me know!

Explorations in Negative Voting

The presentation version I gave of this at MathFest 2009 won me the PME Award for Outstanding Student Research and Presentation.

Or put more formally, this is an elementary construction of the convex regular polytopes of every dimension. This is the presentation I gave at MathFest 2010. For the last part you’ll need some red/blue glasses. These are normally used to create 3d effects, but here we’ll be seeing in 4 dimensions.I plan on making a video out of this at some point that you may enjoy more.

Click Here to See the Presentation

A Platonic Solid is a 3 dimensional object that has the following property: select a face, then an edge on that face, then a vertex on that edge. This selection is called a ‘flag’. All the flags look the same and can be sent to each other by symmetries. This definition is really easily generalizable to work in any dimension: for example, if we’re in 4 dimensions, first you select a 3d cell, then a 2d face, then a 1d edge, then a 0d vertex.

The way to construct these objects is to make them in one dimension lower and try to fit them around a vertex (in the case of 3d, an edge if you’re making 4d things, a face if you’re making 5d things, etc) and ‘fold them up’ into the next dimension. To see if you can fit them, look at their interior angle (that’s the angle listed next to each of the polytopes). Any multiple of them from 3 upwards (you need at least 3 so that when you fold them up they don’t just fold flat) that has total angle less than 360 degrees will work. For example, with the triangle, which has interior angle 60 degrees, you can put 3, 4, or 5 of them around a vertex, making the tetrahedron, octahedron, and icosahedron, respectively.

The last section, with the photos, deals with the Universal Regular Polytopes, the ones that exist in every dimension. The Simplex, on the left, is n+1 (n is the dimension you’re in) points that are equidistant from each other. The Cube, in the center, is the Minkowski Sum of the standard basis vectors-that is, you move one of the vectors along another and look at the shape you trace out. The Cross Polytope, on the right, is the convex hull of the positive and negative standard basis vectors. For the full effect on the 4d versions you should wear red/blue glasses. For the 4d simplex you should see a tetrahedron with another point floating in front of it: all of those points are the same distance from each other. For the cube, the ‘big’ cube is not actually bigger than the ‘small’ cube, it’s just closer. For the cross polytope, the blue lines are in a single 3d hyperplane, while the orange lines traverse the 4th dimension: one point is closer to you, one is further away.

This video does a great job of introducing the concept of a fourth directional dimension. When I talk about dimensions, I’m never talking about time, I’m always talking about directions. Note: I did not make this video, I’m just posting it because I think it gives a good explanation that can hopefully help laymen understand my Convex Regular Polytopes presentation more thoroughly.

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