Archive

Archive for the ‘Geometry’ Category

Fun with Polygonimoes

A lot of nervous energy was building up while I was waiting in the wings at the 2012 TEDxNYED conference.

Luckily, I found some polygons to play around with, so I converted my anxiety into mathematical art.

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Mathematical Image Galleries

This is an great website full of galleries of mathematical images:

http://www.josleys.com/galleries.php

There hundreds of beautiful images in many categories, like fractals, knots, spirals, and tesselations.

There’s even a gallery inspired by the techniques of  M.C. Escher!

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CD Packing Problems

I consider myself an expert arranger of things.  I enjoy rearranging storage space, packing things away, and helping people fill up moving trucks.  It’s a way to apply geometry and optimization techniques; two of my favorite things.

In general, the packing problem entails trying to find the most efficient way to pack a certain kind (or kinds) of objects into a certain fixed space.  Packing probelms are, generally speaking, very challenging because every packing problem is unique.  There isn’t a good, efficient procedure that solves them all.

Here is yet another example of problems with packing problems.  After shedding a bunch of CD cases, I thought I’d try to pack them up in a box.  Here was my first attempt.

I got 49 CDs in the box, but there was a bit of unused space left over.  I couldn’t fit a CD into that unused space, but I thought maybe I could rearrange everything to make some of that space usable.

So I tried again.

The number of CDs in this new arrangement differed by one.  While I can compare which of these packings is more efficient, the problem is comparing all possible packings!  There are a lot of options to consider.

As useless as they are, I ended up having a lot of fun with these CD cases.  I made some parallelepipeds with them and used them to demonstrate Cavalieri’s Principle!

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Folding Website

This is great website from Joseph O’Rourke, author of  “How to Fold It:  The Mathematics of Linkages, Origami,and Polyhedra” .

www.howtofoldit.org

The website has several videos and cool animations that demonstrate some of the basic ideas in mathematical folding, like the one-cut problem, the map puzzle, and folding polygons into convex polyhedra.

There are also a few folding patterns available for download, just in case you’d like to produce a turtle with one cut!

And for more resources on math and origami, check out my fun with folding page!

www.MrHonner.com

Wireframe Contour Maps

I have an adjustable screen for my window, the kind you expand horizontally to fill up the windowsill.  It’s somewhat effective at keeping bugs out of the house.

When it’s not opened all the way up, the two layers of screen overlap in the middle.  Depending on the angle you are looking from, you can see some cool images.

At this angle, for example, I see a contour map of a function of several variables.

I wish I understood where the curves come from!

www.MrHonner.com

Happy 5-13-12 Day!

In honor of today’s date, 5/13/12, I honor one of my favorite triangles:  the 5-12-13 triangle.

Of course, one reason this is a such a nice triangle is because it is a right.  We can easily see that the side lengths satisfy the Pythagorean Theorem

$5^2 + 12^2 = 13 ^2$

Another reason I like this triangle so much is because it plays a part in another of my favorite triangles:  the 13-14-15 triangle!

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This is Not a Rectangle

After having fun exploring rigid and non-rigid frames, I hung one of our indeterminate quadrilaterals up on the board.  The next day, we were proving a theorem about orthodiagonal quadrilaterals, and the final step concluded that a particular quadrilateral was actually a rectangle.

I found a cute little spot to finish our proof.

This elicited a few laughs from students who appreciated the irony.

But apparently, some students in a later class did not appreciate it.  They felt the need to chime in.

As a general rule I must oppose mathematical graffiti, but it’s hard not to respect their position.