My Math Forum  

Go Back   My Math Forum > High School Math Forum > Geometry

Geometry Geometry Math Forum

LinkBack Thread Tools Display Modes
June 16th, 2017, 05:51 PM   #1
Joined: Jun 2017
From: Moon

Posts: 2
Thanks: 0

Polygons in a cube


the situation is the following: A cube is intersected by a plane. Depending on the coordinates of the plane, different polygons can be created by the intersection.
How can I prove (with analytic geometry) that these polygons can be triangles, rectangles, pentagons or hexagons?

Intuitively (and by drawings) I know that this must be the case but how can I prove it? I thought about showing that the sums of the angles are (n-2)*pi for n from 3 to 6 but I didn't manage to calculate the angles with an arbitrary plane.

Does someone have an idea?
tombombadil is offline  
June 16th, 2017, 07:30 PM   #2
Senior Member
Joined: May 2016
From: USA

Posts: 785
Thanks: 312

I do have an idea, but I have not worked it out.

Set up your cube with one corner at the origin of a three dimensional Cartesian grid. You can do this because the orientation of the grid is free with respect to the cube.

Now any plane that is parallel to the xy-plane and intersects the cube will form a rectangle.

Now alter the orientation of the intersecting plane with respect to the xy-plane without making it parallel to the yz-plane or or xz-plane. Presumably there will be only a small number of ways in which such an alteration can be done, and each way should generate a small number of patterns of intersection.
JeffM1 is offline  
June 17th, 2017, 04:41 AM   #3
Joined: Jun 2017
From: Moon

Posts: 2
Thanks: 0

Ok, thanks for the idea!,

I tried to set up the plane like this:
a is the length of the cube,
I chose (a|a|d) as the anchor of rotation, where d is a parameter to go up and down.
Then my plane E is (a|a|d) + r*(0|1|tan(alpha)) + s*(1|0|tan(beta)) where alpha and beta are elements of (0,Pi/2) and r,s the parameters.
Do you think this is correct?

If yes, how can I go on from here?
tombombadil is offline  

  My Math Forum > High School Math Forum > Geometry

cube, plane, polygons

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
How many red cube can surround a blue cube?(all equal size) rnck Art 1 December 5th, 2015 03:49 AM
polygons Chanoch New Users 0 April 28th, 2014 11:19 PM
math question (cube, cube volume, percent) yano Algebra 8 March 25th, 2010 05:40 PM
Self-intersecting Polygons and Simple Polygons telltree Algebra 2 July 28th, 2009 09:28 AM
math question (cube, cube volume, percent) yano Applied Math 4 December 31st, 1969 04:00 PM

Copyright © 2017 My Math Forum. All rights reserved.