3-D Shape That Can Be a Circle, Square, or Triangle in 2-D

Date: 09/16/2006 at 21:50:19
From: Theresa
Subject: Geometry-shapes that pass through and plug other shapes

Is there a shape or object that can plug and pass through three
different holes, a square, a triangle, and a circle?  The same object
must be used to pass through all three.  The dimensions of the holes
are: Triangle has a 6 inch base and a height of 6 inches, Square is 6
inches by 6 inches, and the Circle has a 6 inch diameter.

I have tried every shape that I know of but nothing works.  I would
think it would be a liquid or changeable shape.  Or maybe a sphere of
some sort?

Date: 09/17/2006 at 22:06:43
From: Doctor Douglas
Subject: Re: Geometry-shapes that pass through and plug other shapes

Hi Theresa.

There is a solid object that plugs all three of these holes, and 
passes through each of them as well.  It is shaped like a chisel 
point.

Start by visualizing a cylinder whose height is equal to its diameter:

        
         C           
     .       .           Top view of the cylinder.  O is at the    
   .           .         center.  ABCD is the upper circle, and
   D-----O-----B         WXYZ are located directly underneath these.
   .           .         You can see how this shape will plug a 
     .       .           circle.
         A               
                         
        
         
   D-----A-----B         A     B     C  
   |           |              / \ 
   |           |             /   \ 
   |           |            /     \
   |           |           /       \
   |           |          /         \
   Z-----W-----X         W-----X-----Y
         

The side view of the cylinder appears as a square, because the 
diameter AC is equal to the height AW.  Now, imagine truncating the 
solid by planes BDW and BDY, leaving the triangle shown in the lower 
right.  The solid that remains will pass through and plug a circle 
(its base WXYZ is a circle), a square (using square DBXZ), and a 
triangle (BWY) whose height is equal to its base.  

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/