Working Algorithm for Camera Problem

This is a working algorithm for the camera problem, this algorithm works with polygons (this works for any polygon with these features in the worst case scenario) with a finite amount of equal straight sides, a finite amount of straight protruding walls inside the room and a finite amount of concave angles. The algorithm goes as thus.

a=concave angles         w=protruding walls         c=camera

                                           a+(w+1)= c
this excludes polygons with no protruding walls or concave angles which will always need 1camera, polygons with 1 concave angle will always need 1 camera also.

A shape we didn't discuss much in class is the Hyper cube. A Hyper cube is hard to explain because it is in the "Fourth Dimension." What are YOUR thoughts on the Hyper cube? Please leave comments below. Above is a 3d representation of a Hyper Cube.

Cool problem I've been working on:
You have five shapes on a two dimensional plane. Can they all touch each other? (With our definition of touch being sharing more than one points as borders) No overlapping, no third dimension.

Just to clarify, they can be any size, any shape. They don't have to be congruent (they can be 5 completely different shapes)

I think I have a proof of this not being possible, though its not something I can express through words. Not even sure if the proof is adequate. Eager to hear your thoughts.

The hardest one that we actually tried to discuss in class was probably 'shape' (it's probably for the best that we didn't touch the hypercube). That said, it was also my favorite to discuss, simply because not everybody agreed on what a shape is supposed to look like, and so the argument went beyond phrasing and semantics.

polygon

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain or circuit.

I don't really have a single shape or concept that I enjoyed talking about I just really like the debates about definitions and the fact that their are always "monsters" to create more problems and things to be solved and it's sortof the ongoing work and debate that goes into every temporarily finished deffinition.

One object (geometric) that obviously was hard for us to define was point, because we had to first define all this other stuff that we really did not need to, because some people are really picky about their definitions. Cough cough. What do you guys think? Is this a point? 

Equality of shapes

Shapes can be considered the same or equal when they have all the same properties and have the same angles and length of sides.

A Strange Polygon

I don't know what a polygon is anymore! I thought that I knew, but it turns out I don't. I mean, a square is a polygon, and so is a triangle. But what about this thing?

Our glossary definition says that a polygon is "a closed 2D shape that has multiple straight (not curved) lines, where every line is touching another line on both ends." And this shape obeys that definition. But it has an infinite number of sides! It also has an inside and an outside, which I guess is important.

What do you think? Is this a monster, or just an unusual example of a polygon?

Welcome!

Hi, and welcome to our class site! You'll find lots of resources here to learn about geometry and to participate in our class. Soon (hopefully) you'll also be able to share your work here, too.

Feel free to check out the various pages I've set up. They're not all functional yet, but we'll get there!

Just to test out posting pictures, here's a photograph of the Yoshimoto cube that Paul Salomon and I made over the summer:

I'm looking forward to a great year ahead!