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Topic: The Solid Spheres of Cubic Fury. (Read 1899 times) |
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rloginunix
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The Solid Spheres of Cubic Fury.
« on: Oct 2nd, 2014, 10:43am » |
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The Solid Spheres of Cubic Fury.
Edmund graduated from college with high marks in exact sciences while really sucking at humanities. Still, as a graduation present, his grandma decided to give him a beautiful Chinese vase from the Qin dynasty era. The vase is a perfect sphere of a large and yet unknown radius.
The family lore has it that Edmund's distant ancestor saved the life of a Chinese Master Potter who, were it not for Edmund's brave forebearer, would have drowned in the treacherous waters of a fast and ruthless mountain river he was trying to cross in order to get to the local bazaar where he was planning to sell his pottery. The vase was a Master Potter's token of gratitude.
Exactly how Edmund's ancestor brought the vase home is a story shrouded in mystery but now it's his job. Grandma's stipulation - Edmund must transport the vase from her house to his house himself.
Edmund notices a box - a, Platonic Solid, Hexahedron with a side length of a. However, the box is a bit too big for the vase. There is a chance that the vase might break while en route. Quick thinker that Edmund is - eight identical styrofoam balls of proper radius placed one per corner of the box will hold the vase tight.
Using a compass Edmund draws circles equal in number to the fifth prime and using a straightedge and a pencil he draws straight lines equal in number to the second prime and determines the radius of the vase, R.
1). What is Edmund's procedure to determine R?
2). What is the radius of the (solid) cushion spheres, r?
3). How did Edmund's ancestor bring the vase home?
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rloginunix
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Re: The Solid Spheres of Cubic Fury.
« Reply #1 on: Oct 3rd, 2014, 6:50am » |
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Hint: Edmund drew first four circles on the sphere.
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dudiobugtron
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Re: The Solid Spheres of Cubic Fury.
« Reply #2 on: Oct 3rd, 2014, 1:31pm » |
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re: your hint, do you mean On the actual vase? 
PS:, so I don't have to 'work' these things out later:
A hexahedron is a cube, the 5th prime is 11, and the 2nd prime is 3. So he draws 11 circles and 3 straight lines.
PPS: is the answer to number 3:
On the river?
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| « Last Edit: Oct 3rd, 2014, 1:34pm by dudiobugtron » |
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rloginunix
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Re: The Solid Spheres of Cubic Fury.
« Reply #3 on: Oct 3rd, 2014, 2:40pm » |
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on Oct 3rd, 2014, 1:31pm, dudiobugtron wrote:| re: your hint, do you mean On the actual vase? |
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Yes.
on Oct 3rd, 2014, 1:31pm, dudiobugtron wrote:PS:, so I don't have to 'work' these things out later:
A hexahedron is a cube, the 5th prime is 11, and the 2nd prime is 3. So he draws 11 circles and 3 straight lines. |
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That is correct.
on Oct 3rd, 2014, 1:31pm, dudiobugtron wrote:PPS: is the answer to number 3:
On the river? |
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It can be. This was just a humorous question to see if the reader is paying attention since the problem statement contains the answer - "a story shrouded in mystery",  . But, again, any scientifically plausible explanation will do and is welcome.
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dudiobugtron
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Re: The Solid Spheres of Cubic Fury.
« Reply #4 on: Oct 5th, 2014, 11:40am » |
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I'd thought that, once we had discovered the answers to 1 and 2, it would give us some insight into 3. For example, if we discovered that the vase was actually 100m in radius, it would limit the options considerably!
This is a cool riddle btw; one I'm not sure I have the technical know-how to solve, but one that is really fun to think about nonetheless.
on Oct 3rd, 2014, 2:40pm, rloginunix wrote:So, his grandmother gets him a priceless antique heirloom vase and entrusts him with its safe journey to his house, and he goes and draws all over it?
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rloginunix
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Re: The Solid Spheres of Cubic Fury.
« Reply #5 on: Oct 5th, 2014, 1:35pm » |
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on Oct 5th, 2014, 11:40am, dudiobugtron wrote:| So, his grandmother gets him a priceless antique heirloom vase and entrusts him with its safe journey to his house, and he goes and draws all over it? |
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That is the crux of the procedure. Once you know what it is it is actually not that bad because:
Hint: it is not full circles you are after - only their interesting intersection points of which two pairs of intersecting circles will produce how many? But you do not need all of them.
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rloginunix
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Re: The Solid Spheres of Cubic Fury.
« Reply #6 on: Oct 6th, 2014, 8:20pm » |
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Pick two points O1 and O2 on a sphere at random, for practical purposes not too far from and not too close to each other. By definition O1 and O2 sit on a great circle, say, GC1.
Using O1 and O2 as poles draw two intersecting circles of equal radius, R1, to obtain two intersection points P1 and P2 which (by definition) also sit on a great circle, GC2 (orthogonal to GC1).
Repeat. Using O1 and O2 as poles draw two more intersecting circles of equal radius, R2 <> R1, to obtain two more intersection points P3 and P4 which (by definition) also sit on GC2. Four circles drawn so far.
However, you need only three of these points ...
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rloginunix
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Re: The Solid Spheres of Cubic Fury.
« Reply #7 on: Nov 9th, 2014, 2:17pm » |
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Tying the loose ends, again, before I forget.
A distance between any 3 such points, measured with a compass, is linear or planar - imagine a gopher chewing his straight line tunnel through the sphere from one point to another. Carrying 3 such distances on to a piece of paper on a table allowed Edmund, an street dog he truly is, to construct a triangle, 1 straight line, 3 more circles, the radius of whose circumcircle is the radius of the sphere. To construct the circumcircle Edmund used 4 more circles, 11 total, and 2 more straight lines, 3 total.
Now that the radius of the vase, and the size of the cube, are known it should be really easy to calculate the radius of the smaller cushion spheres.
Lastly, Barukh and Deedit beat me to it by only about 9 years but I think I can add a fresh new prospective to the old puzzle.
Several years later Edmund received his Ph.D. in math and a rowdy crowd, celebrating the event at Edmund's house, accidentally knocked the vase off the stand, the vase fell on the floor and broke into pieces. How unfortunate! To submit a claim to his insurance company Edmund needed to calculate the exact radius of the vase again.
He picked up a reasonably sized fragment of the vase off the floor and managed to find the radius of the vase nonetheless. How did he do it?
Click for the answer.
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