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An 'Impossible' Solid (Posted on 2004-10-23) Difficulty: 4 of 5
The discipline of Draughting/Drafting usually has exercises requiring the presentation of 3 elevations of an object; aerial or plan view, front view and side or end view. A standard house brick would be 3 rectangles drawn in relation to its dimensions.

I understand that somewhere through the 1930ís a German architect proposed a drawing for a solid object which many deemed impossible, but I have a lovely brass model that invalidates those claims.

The challenge was: Given one drawing that represents all three elevations - Create the object!

Examples: A square is a cube. A circle represents a sphere but a circle crossed with a ' + ' sign might be a beach ball with circles around its 'x,y,z' circumferences; like an orange cut into 8.

NOW, this object in question is represented by a circle crossed by an 'X' or multiplication sign.

MY CHALLENGE is twofold:
1. What does this object look like? Describe as many of its properties as possible.
2. How might you create it as a demonstration in, say, 2 or 3 minutes? I suggest a firm but pliable medium like children's 'playdough' and a tool like a very simple kitchen utensil would reasonably create an approximation of this solid.

See The Solution Submitted by brianjn    
Rating: 3.2000 (5 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(4): Finally, I see it. | Comment 22 of 25 |
(In reply to re(3): Finally, I see it. by brianjn)

I wish I had thought of the intersection of three cylinders.  That works perfectly.

Description:
Imagine a cube that has all its faces covered with square pyramids.  The tips of the pyramids are rounded off like a circle.  Where the pyramids meet each other, the edges blend in arcs.  The X seen is made up of the visible edges of a pyramid.  The circle seen is made up of the round tips of the 4 adjacent pyramids, seen from a profile.  Of course, the solid is symmetrical, so all elevations look the same.

This solid has 12 faces, each face being a square that is bent in a 90 degree arc, from one corner to the opposite corner.

Demonstration:
I'm not too artistic, so I was unable to incorporate any cookie cutters or utensils or anything of that sort.  However, I managed to render this 3-d object and rotate it much in the same way I rendered my avatar.  Unfortunately, I don't have a good way of displaying a picture of it, and besides, a single snapshot of it would not work nearly as well as a video of its rotation.  It took much longer than 2-3 minutes.

I don't suppose anyone is interested (I wouldn't be), but I have the equations I used here. 

Notes: About 2000 points are graphed (x,y) with different values of i between 0 and 1.  The second group of equations essentially just rotate the object rendered in the first group of equations, as well as changing the color.  t, t1, and t2 are time variables that I can make increase at a set rate.  $PI represents pi.

p=1000
q=i*p*2/$PI+.5;
d=min(  1/cos( (q-floor(q)) * $PI/2 - $PI/4) /2 * sqrt(1-sqr(2*i-1))  , .5  );
x=sin(i*p)*d;
z1=cos(i*p)*d;
y=i-.5;

y1=sin(t)*y+z1*cos(t);z2=cos(t)*y-sin(t)*z1;
y=sin(t1)*y1+x*cos(t1);x1=cos(t1)*y1-sin(t1)*x;
x=sin(t2)*x1+z2*cos(t2);z=cos(t2)*x1-sin(t2)*z2;
c=z/2+.5;
green=c;red=-c+1;blue=sin(t3/4)/2+.5;


  Posted by Tristan on 2004-11-18 01:29:23
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