Sliced cube

[bonanova]

You slice a cube with a plane and examine the new surface you've made.
Is it possible for this surface to be

• an equilateral triangle?
• a square?
• a regular pentagon?
• a regular hexagon?
• a regular polygon with more than 6 sides?

Give intuitive reasons.

[Guest]

1) yes, any of the corner, make sure is equalaterial when you slice in

2) Yes, any straight cut follow x-y plane, y-z plane or z-x plane.

3)...still thinking... <_<

[Guest]

You slice a cube with a plane and examine the new surface you've made.

Is it possible for this surface to be

1. an equilateral triangle?
   
2. a square?
   
3. a regular pentagon?
   
4. a regular hexagon?
   
5. a regular polygon with more than 6 sides?
   

Give intuitive reasons.

square is the most straight forward just cut parallel to a side.

triangle is also easy by cutting on one corner such that the cutting plane passes three edges.

regular hexagon is also possible if we cut it at an angle such that the cut passes one surface then two edges which are on the diagonal and then the opposite surface.

pentagon is possible but not regular

a polygon with more than six sides is not possible.

[Guest]

3) Yes, starting from top surface of 1 corner (top right most and inner most) , cut diagonally downwards but above the corner of (bottom left most and outermost) will give you regular pentagon,

4) Yes, from top surface, follow the line where length/2 x length /2 can cut diagonally downwards until you meet the line where length/2 x length x2 at the bottom surface.

5) Nope because my thinking has run dried!

Sorry for my poor english explanation

[Guest]

Darn, Imran beat me to it.

You can't have a polygon with more than six sides by slicing a cube with a plane becuase the cube only has six faces. The only way you could do that is if we were to warp the plane or the cube.

[Guest]

Darn, Imran beat me to it.

You can't have a polygon with more than six sides by slicing a cube with a plane becuase the cube only has six faces. The only way you could do that is if we were to warp the plane or the cube.

Good explanation I guess. I could visualise it but not explain it .

But lets see what bonanova says

[bonanova]

Good explanation I guess. I could visualise it but not explain it .

But lets see what bonanova says

Results so far:

Triangle - Yes=2; No=0. - good reasons given by several.

Square - Yes=2; No=0. - pretty self evident

Pentagon - Yes=2; No=0. - imran says possible, but not regular; woon gives a method.

Hexagon - Yes=2; No=0. - with a nice description of how by woon

Seven+ - Yes=0; No=3. - proof given by Jaod997.

Does everyone agree with the consensus?

Anyone care to venture other reasons for [or against] the pentagon?

Try to do this using logic and symmetry, and not with sketches.

[Guest]

I can see the pentagon, but I don't see how it would be a regular pentagon. If you cut from one of the top corners the sides that are higher up will have to run the entire length of the faces that come off of that corner, while the three sides on the bottom can't possibly be that long since you have to basically cut them short in order to get 5 sides.

Edit for this quick plea: Once the discussion part is over, I would love to see some sketches. Especially the pentagon.

Edited June 10, 2008 by ChuckJerry