tetrahedron... how many right angles can the 4 surface-areas have?

hmm_marie · 2006-11-20 10:32:51

there are 4 triangular surfaces of a tetrahedron. how many right angles can they have?

there can obviously be none, right? in a regular tetrahedron, for example...
there can be 4 at the very most, because every surface can only have one right angle as the sum of the angles has to be 180° and therefore no 2 right angles can exist in a triangle.

but how do i show that there can be 1, 2, 3 and 4? is that possible?

that only 1 triangular surface can have a right angle seems quite plausible to me... but i can't prove it.

i think 3 right angles in the same corner of the tetrahedron are possible too. but i can't prove it, as I don't know how to show that the other triangle can't possibly have a right angle.

could anyone please help me on this one?

John E. Franklin · 2007-08-13 04:14:20

3-D geometry is very hard to imagine, and the formulas are in general still over my head.
There is an interesting thing to try to imagine on a tetrahedron, which is why
I just searched up this old post, to stay on a relevant topic.
There are 3 pairs of opposite perpendicular edges to a tetrahedron.
And if you try to make pairs of separated planes out of these pairs of
opposite edges, you might run into a quandrum in your mind.
I see that you have 3 pairs of planes that are most probably
not going to form a cube. However, they must form some impossible
6-sided shape, with impossibly shaped sides, or yet undetermined to me at least.
But the awesome part is, is that the bizarre shape must be
quite symmetrical, as it was constructed directly from the perfect
tetrahedron...
(Here's some neat pictures maybe a little off the subject, but gets the mind thinking.
http://gregegan.customer.netspace.net.au/SCIENCE/KleinQuartic/KleinQuartic.html)

Last edited by John E. Franklin (2007-08-13 04:32:50)

mathsyperson · 2007-08-13 05:42:31

Hang on. I'm not getting how you can define a plane with two lines (unless they're coplanar).
How would those pairs of lines from the tetrahedron form planes?

John E. Franklin · 2007-08-13 10:43:23

Yes mathsy, I made it out of my Dad's modeling clay!!
It's a cube!! It's a cube.
Here's a picture.
And if you lines are perpendicular but not in the same plane, then
you just make two planes line a capacitor.
Here's the picture I took of it touched up with lines.
The cube is on the outside made of paper held on with
tooth picks.

John E. Franklin · 2007-08-13 11:19:06

I just started googling on "tetrahedron in cube", and found a few drawing of this and calculations.
So it's nothing new. Just new to me!! I also took my rubiks cube and drew diagonals on it that alternate zigzaging around the cube and guess what it makes?? Try it!! It's a ball.
You can take a rubiks cube and a ruler and permanent marker, and start drawing a diagonal on one side.
Then connect the ends of this diagonal over the corners to the other two sides and draw diagonals on them too, so it could be done without lifting the pen, except that you go in two directions.
But there are lots of ways to proceed. But anyway. Get an icecube and try to paint the diagonals on it with something. I'd like to know what ink sticks to an icecube!

Last edited by John E. Franklin (2007-08-13 12:08:01)

Math Is Fun Forum

#1 2006-11-20 10:32:51

tetrahedron... how many right angles can the 4 surface-areas have?

#2 2007-08-13 04:14:20

Re: tetrahedron... how many right angles can the 4 surface-areas have?

#3 2007-08-13 05:42:31

Re: tetrahedron... how many right angles can the 4 surface-areas have?

#4 2007-08-13 10:43:23

Re: tetrahedron... how many right angles can the 4 surface-areas have?

#5 2007-08-13 11:19:06

Re: tetrahedron... how many right angles can the 4 surface-areas have?

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