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Thinking in 3D

Nanjangud Nanjundaiah
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Joined: May 15, 2001
Posts: 186
How good are you in visualizing in 3D? Do you feel all your thoughts lack the depth? Do you want to experience the pleasure of thinking and imagining in 3D? Then here's the right puzzle for you!
I will be providing views (top, front etc) of a solid object I have in mind. You will have to visualize and construct this object in your mind, and give me the 3D view, more precisely, the isometric view.
Here's a simple example:
A cube has all the 6 views (top, bottom, left, right, front, hind) the same: It's a square. The isometric view is on the right.<pre>
-- / \
| | |\ /|
-- \ | /</pre>
Get the picture? See this applet for fuller understanding.
Now, here's my puzzle. All the six views of my object look like this:<pre>
-----
|\|/|
-----
|/|\|
-----</pre>
That's like the flag of Great Britain, except that this is a square.
Now can you imagine how the real solid would look like? If you think you have the answer, try carving it into a piece of soap. (But be prepared to hear your significant other screaming, "Where's the soap gone!")
Map, do you see, mysteriously, there are triangles involved here

[This message has been edited by Nanjangud Nanjundaiah (edited June 04, 2001).]


The Rancher Formerly Known As Nanhesru Ningyake.
Greg Harris
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Joined: Apr 12, 2001
Posts: 1012
i am curious; after i figure it out and carve my answer, how do i post it to this forum???


what?
Junilu Lacar
Bartender

Joined: Feb 26, 2001
Posts: 4745
    
    7

Originally posted by Nanjangud Nanjundaiah:
Now can you imagine how the real solid would look like? If you think you have the answer, try carving it into a piece of soap.

But I use liquid soap...
Didn't need to carve anything anyway...just think inside the box this time. Carving that in soap would be pretty challenging, too.
Here's a hint (hidden for those who prefer to solve it themselves):
You'll have better luck cutting and pasting together pieces of paper or cardboard.

Junilu

[This message has been edited by JUNILU LACAR (edited June 04, 2001).]


Junilu - [How to Ask Questions] [How to Answer Questions]
Jim Yingst
Wanderer
Sheriff

Joined: Jan 30, 2000
Posts: 18671
Yeah, that's a challenge in itself. Maybe we can post applets to display our solutions. I have a solution, but I'm not sure it's the only one possible. Though it does have a certain elegant simplicity - if there are other solutions, I suspect that they are much more odd-looking and asymmetric.
Here's my solution:
000-011-110
000-011-101
000-110-101
002-011-112
002-011-101
002-112-101
020-011-121
020-011-110
020-011-110
022-011-121
022-011-112
022-121-112
200-211-101
200-211-110
200-101-110
202-211-101
202-211-112
202-101-112
220-211-121
220-211-110
220-121-110
222-211-121
222-211-112
222-121-112
Of course, you have to decode it first.


"I'm not back." - Bill Harding, Twister
Nanjangud Nanjundaiah
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Joined: May 15, 2001
Posts: 186
>after i figure it out and carve my answer, how do i post it to this forum
Greg, you can create an applet, or simply draw the isometric view, using Paint Shop Pro or something better, create a .jpg and post it onto your homepage.
>Carving that in soap would be pretty challenging, too.
Yep, I have tried this... you need some extreme patience. I eventually created a model with a stick frame and paper surfaces.
>I have a solution, but I'm not sure it's the only one possible
I think there can be only one solution. This is because there are no hidden lines. (Dunno how I can prove that the solution is unique though. I will leave that to you Jim )
I "created" this solid when I was in school; I was reading a Physics textbook explaining crystal lattice structures and this suddenly occurred to me. If it's really undocumented let's call it the Ningyake crystal (something like "Bucky Ball" )
Jim, may I know how to decode your solution? I think you are providing spatial coordinates for all the points...

[This message has been edited by Nanjangud Nanjundaiah (edited June 04, 2001).]
Peter Lyons
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Joined: Sep 12, 2000
Posts: 202
The solution I drew on paper is basically an eight-pointed star.
Jim Yingst
Wanderer
Sheriff

Joined: Jan 30, 2000
Posts: 18671
For my solution, each line represents a surface of the solid - e.g. "000-011-110" represents a triangle connecting points (0, 0, 0), (0, 1, 1), and (1, 1, 0). I just didn't feel like including all those parentheses and commas. But yeah, you can basically view it as an eight-pointed star of sorts. It shouldn't be too difficult to carve - just start with a cube, and take a wedge out of each edge.

[This message has been edited by Jim Yingst (edited June 05, 2001).]
Greg Harris
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Joined: Apr 12, 2001
Posts: 1012
i folded it in paper yesterday... i came up with the 8 pointed star as well. i just was not sure if the picture above was supposed to show the middle section convex or concave... so i went with convex.
i was originally a mechanical engineering student (10 years ago) and spent some time drawing flat layouts of 3D objects, then cutting them out, folding and pasting the tabs together... i could not believe i was getting a grade to play!
Junilu Lacar
Bartender

Joined: Feb 26, 2001
Posts: 4745
    
    7

Originally posted by Jim Yingst:
> It shouldn't be too difficult to carve - just start with a
> cube, and take a wedge out of each edge.
I had originally envisioned a bunch of intersecting quadrilateral planes that made a kind of triangular honey-comb thing, which seems to be another solution. That would be really hard to do on a piece of soap. Pretty challenging to do on paper too. I think AutoCad would have better luck rendering it in 3D.
Junilu
Nanjangud Nanjundaiah
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Joined: May 15, 2001
Posts: 186
>For my solution, each line represents a surface of the solid
How on earth did you generate this?! I tried drawing a grid and marking out the lines - it started to look like a sheet-metal object than a solid - so are you certain that this is a real solid?
>i came up with the 8 pointed star
This is actually a 6-pointed star, so I wonder where you are getting the extra points...
>i could not believe i was getting a grade to play!
Wasn't that fun? I loved engineering drawing too - especially creating perspective views (more fun than isometric!)
>That would be really hard to do on a piece of soap.
It will help to have your soap dry, and not squishy wet from the shower
Greg Harris
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Joined: Apr 12, 2001
Posts: 1012
Originally posted by Nanjangud Nanjundaiah:
i came up with the 8 pointed star
This is actually a 6-pointed star, so I wonder where you are getting the extra points...

oops, i meant 6. i actually only folded one side of the cube, so i was not paying attention when i added up the points in my head.
Jim Yingst
Wanderer
Sheriff

Joined: Jan 30, 2000
Posts: 18671
>> For my solution, each line represents a surface of the solid
> How on earth did you generate this?! I tried drawing a grid
> and marking out the lines - it started to look like a
> sheet-metal object than a solid - so are you certain that
> this is a real solid?
What, you doubt me? I'm hurt. Actually I basically did this in my head, with aid from some sketches I made, so it's certainly possible that an error crept in somewhere. But I'm sure that the solid in my head is a real, closed solid. What's wrong with sheet metal, provided it eventually closes?
Think of it this way. Start with a solid cube with corners at 000, 002, 020, 022, 200, 202, 220, 222. Now slice into this from 000 to 011 and 101. Make another slice from 002 to 011 and 101 (i.e. to the same points the last slice extended to, but from a different angle. You have now removed a wedge-shaped region from the adge that connected 000 to 002. Do the same for the other 11 remaining edges. The resulting solid should be symmetric about the planes x = 1, y = 2, and z = 1. Here's a top view - origin is at lower left corner; x axis goes to the right; y axis is up; z axis comes out of the screen.
<pre>
^ Y
|

|
022 121 222
+----+----+
|\ | /|
| \ | / |
| \ | / |
| \|/ |
011+----+----+211
| /|112 |
| / | \ |
| / | \ |
|/ | \|
- - +----+----+ - - -> X
002 101 202
|
|
</pre>
Each plus sign is a point, marked with its coordinates. E.g. "202" means the point (2, 0, 2). Pay particular attention to the z coordinates - the corners and center are all at the same height z = 2, while each edge point is at height z = 1. Of course there are also four corners at 000, 200, 020, and 020, as well as one more center point at 110, which are not visible because they're on the other side of the cube.
>> i came up with the 8 pointed star
> This is actually a 6-pointed star, so I wonder where you are
> getting the extra points...
Although Greg has subsequently recanted, I am sure that my solution does have eight points, not six. The eight points are the eight corners of the original cube, at 000, 002, 020, 022, 200, 202, 220, 222. They are made even more pointy by the wedges that have been removed from each edge. Perhaps we actually have different solutions? Can someone give a representation of this solution that looks like a six-pointed star?
[This message has been edited by Jim Yingst (edited June 06, 2001).]
Greg Harris
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Joined: Apr 12, 2001
Posts: 1012
my object looks like this...
Click here...
maybe i am crazy, but that is what i came up with. this is only half, so you have to put another on the back of this one.
but, now that i see your idea, i like it better, jim. mine is probably too simple.
Peter Lyons
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Joined: Sep 12, 2000
Posts: 202
Mine is definetely an 8 pointed star.
I made a quick drawing in Paint but can't figure out how all you geeky computer people include an image with your post...
So, to describe it again, it's two concentric pyramids, with the second one oriented so its corners protrude from the center of each triangular face of the first pyramid.
(someone tell me how to add a .bmp file!)
Greg Harris
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Joined: Apr 12, 2001
Posts: 1012
it first has to be available on the web. i just posted mine on my company website... i will be happy to put yours on there as well if you want to email it to me.
greg_harris@acm.org
however, some of the real computer geeks have other ways to get around this. i am not quite worthy of full geekdom, yet.
Greg Harris
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Joined: Apr 12, 2001
Posts: 1012
here is Peter Lyons attempt...
Click here...
it is on the same site as my attempt listed above.
Jim Yingst
Wanderer
Sheriff

Joined: Jan 30, 2000
Posts: 18671
Both links go to the same url, http://www.daveharrisrealty.com/greg/misc/ . This goes to the picture of Greg's folded paper star on a blue laptop. I think we need a different URL for Peter's solution.
Greg Harris
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Joined: Apr 12, 2001
Posts: 1012
they are both on the same page for now... just hit "refresh" or "reload" and peter's should come up. i will create a new page tomorrow after class.
Nanjangud Nanjundaiah
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Joined: May 15, 2001
Posts: 186
YES!
Closed!

Jim, I started drawing all the lines (thanks for the detailed explanation, I was using the same technique ), and when your solid began to resemble Peter's diagram, I knew that you'd got it!
Greg, the paper model is correct if the end points aren't bent down, that's when it would resemble the top part of Peter's diagram... five more of those, and you have the full solid
Peter, that's beautiful. You have the (most easily verifiable ) answer. So what was I saying about a 6-pointed star? I was talking about the flat isometric diagram, and when you said 8-pointed star, you were referring to the 8-pointed solid!
Wasn't that fun!

[This message has been edited by Nanjangud Nanjundaiah (edited June 06, 2001).]
Eager Beaver
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Joined: Feb 26, 2001
Posts: 187
Hi all,
In the same vein, front and top views of an object are a square within a square with centres coinciding. Figure out the shape of the object.

[This message has been edited by Eager Beaver (edited June 07, 2001).]
Peter Lyons
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Joined: Sep 12, 2000
Posts: 202
I have a new drawing for Mr. Beaver which Greg says he'll post later. For now, a rough side view describes it pretty well:
<pre>
|\
| \
| \
| \____
| \ |
| \ |
| \|
| \
| \
| \
-----------
</pre>
from the front or top it would appear as two concentric squares, one smaller than the other, both oriented the same way.
NAH! It doesn't display my spaces, so that doesn't look right at all. Wait for the .GIF
[You needed <PRE> tags to preserve the spaces with all characters the same size; I have inserted them for you - Jim]
[This message has been edited by Peter Lyons (edited June 07, 2001).]
[This message has been edited by Jim Yingst (edited June 07, 2001).]
Nanjangud Nanjundaiah
Ranch Hand

Joined: May 15, 2001
Posts: 186
Is it something like this:
<pre>
------------
\ \
|\ ----- \
| \ \ |\ \
| \ \ | \ \
| \ ----- \
| \ \
| ------------
\ | |
\ | ------ |
\ | | \ | |
\ | | \| |
\ | ------ |
\| |
------------
</pre>
Peter Lyons
Ranch Hand

Joined: Sep 12, 2000
Posts: 202
Yeah, Nanjangud's works too.
Nanjangud Nanjundaiah
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Joined: May 15, 2001
Posts: 186
Actually Peter, you are more correct
Mine would have hidden lines in it - yours won't!

[This message has been edited by Nanjangud Nanjundaiah (edited June 07, 2001).]
Greg Harris
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Joined: Apr 12, 2001
Posts: 1012
sorry for the delay... Peter Lyons sent this to me while i was in class today at 11:30 EST.
Click here...
Eager Beaver
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Joined: Feb 26, 2001
Posts: 187
Hey Peter & Greg ('Lyons & Harris' Inc),
You have squared in on those squares!! The 3-d rendering was wonderful. Nanjangud, I appreciate your patience in belabouring to put that block in your post. I myself could not post a proper image to explain my puzzle....everytime I did the sides of the inner square would collapse.
SOLVED
-EB.
Greg Harris
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Joined: Apr 12, 2001
Posts: 1012
i don't get any credit here... i just posted the picture to my website! my 6 pointed star failed miserably.
Nanjangud Nanjundaiah
Ranch Hand

Joined: May 15, 2001
Posts: 186
Originally posted by Greg Harris:
my 6 pointed star failed miserably.

No you didn't. It was pretty much on the right track. Cheer up
 
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subject: Thinking in 3D