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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 1930’s 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.

  Submitted by brianjn    
Rating: 3.2000 (5 votes)
Solution: (Hide)
Building the model is probably the best way to go, even if mentally, so you may need to visualise quite strongly; er .. and consider it within an ‘xyz ‘ environment.

Using a circular cookie/biscuit/scone cutter [or something of that nature] cut a cylindrical form that is longer than the cylinder’s diameter.

At 90 Deg to that cylinder’s axis, cut another cylinder. A third cut is now needed in the 3rd dimension.

A cylinder has been created and trimmed in the 2nd and 3rd dimensions (or elevations).
The object has the initial appearance of a sphere but with a subtle resemblance to a cube.

Properties:
The object has 8 vertices (as does the cube). [IMPORTANT - See NOTES below]. All surfaces are curved cylindrically. There are 12 of these. Each dimension has 4. (On a cube there are 12 edges).

The ‘X’s have their endpoints at the vertices. There are 6 of these (the cube has 6 faces). Between adjoining surfaces the edges are sharp at the vertex but fade into oblivion at the midpoint of each ‘X’.

The vertices remain the same but the values of edges and faces are inversed.

QUESTION ( as an aside, but a thought): Disregarding the ‘Plan/Elevation’ stipulation of this submission, what forms do: eg, a tetrahedron, dodechahedron, … take if we incur a similar inversion process?

NOTES from Charlie. He has passed me these notes which I accept. Note Vertices.
Actually the object has 14 vertices: 8 where 3 of the surfaces meet, corresponding to the vertices of the cube, and 6 where 4 surfaces meet, corresponding to the faces of the cube.
The cylindrically curved faces correspond to the edges of the cube, and if flattened out would form a rhombic dodecahedron.
The extra 6 vertices are the midpoints of the X's, and though the edges "fade into oblivion" there, it's still a vertex. Like the apex of a vaulted ceiling where perpendicular cylindrical surfaces meet.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re(3): Finally, I see it.brianjn2009-11-09 21:49:15
QuestionNaive (?) questionFrankM2008-03-13 11:12:29
re(5): Finally, I see it.brianjn2004-11-18 02:18:39
re(4): Finally, I see it.Tristan2004-11-18 01:29:23
re(3): Finally, I see it.brianjn2004-11-16 19:41:29
re(2): Finally, I see it.SteveH2004-11-16 17:41:32
re: Finally, I see it.brianjn2004-11-15 22:10:08
re: Finally, I see it.owl2004-11-14 11:08:02
SolutionFinally, I see it.SteveH2004-11-12 17:40:54
re(2): Review - lovely thoughts - owl's modifiedbrianjn2004-11-12 00:37:11
Solutionre: Review - lovely thoughts - owl's modifiedSteveH2004-10-31 11:51:00
Review - lovely thoughtsbrianjn2004-10-26 07:09:41
re(2): Maybe Solutionowl2004-10-24 15:47:24
Solutionre(2): Solution - fixedTristan2004-10-24 15:19:46
re: Maybe SolutionSteveH2004-10-24 14:39:15
Hints/Tipsre(2): X marks the spotbrianjn2004-10-24 06:21:05
SolutionMaybe Solutionowl2004-10-24 03:52:55
re: Solution - flawTristan2004-10-23 22:29:10
re: X marks the spotSteveH2004-10-23 22:02:58
re: Possible solutionTristan2004-10-23 21:15:57
SolutionSolutionTristan2004-10-23 21:01:13
Some ThoughtsX marks the spotLarry2004-10-23 18:59:35
re: Possible solutionSteveH2004-10-23 18:03:05
Hints/TipsPossible solutionSteveH2004-10-23 17:45:22
No Subjectbob9092004-10-23 17:31:55
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