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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.)
Solution re(2): Solution - fixed | Comment 12 of 25 |
(In reply to re: Solution - flaw by Tristan)

As I said in my last comment, my solution has arcs that when seen from the side, will extend part of the X slightly beyond the circle, and it extends different parts of the X on different elevations.  To solve this, I will simply extend all the sticks slightly so that all parts of the X are extended beyond the circle.  That way, at least it is symmetrical, and all elevations look the same.

So repeating my solution in full:

  A--B  Sticks CDGH are seen from front,
/| /|  DBHF from side,
C--D |  and ABCD from top.
| E|-F  Stick E is not needed, and neither is D.
|/ |/  

Using the diagram above of a cube, imagine the center of the X as the center of this cube.  There are 6 sticks going from the center to points C, A, B, F, H, and G.  Make 6 arcs between these points (not quite at the tips of the sticks), and there you have the solid, or rather in this case, more of a bent circle.

Looking from an angle between all three elevations, the solid should look like a circle with a * crossing it.  When rotating to any of the elevations, two of the arcs will disappear, not going behind the circle, but simply rotated to a position where only a line is seen.


To create this figure--in fact, I did this myself just a while ago--cut out a circle and a half from some paper.  In the circle, cut out a 353 degree arc, and in the semi-circle cut out a 71 degree arc.  Fold the 353 arc into 5 equal arcs (the exact arc size should be about 70.5 degrees).  Tape the arcs together, and manipulate the 423 degree bent circle into the figure described above.  Of course, this model is not perfect--the sticks don't extend beyond the circle like they should, and I think the arcs aren't supposed to have a radius equal to the length of the folds.


  Posted by Tristan on 2004-10-24 15:19:46
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