Re: psychology of geometric visualization
From: Chan-Ho Suh (suh_at_math.ucdavis.nospam.edu)
Date: 07/07/04
- Next message: Eric Gisse: "Re: A little lesson for sqrt(144) year olds."
- Previous message: Michael N. Christoff: "Re: Topology and Dynamical Systems"
- In reply to: Elaine Jackson: "psychology of geometric visualization"
- Next in thread: Herman Jurjus: "Re: psychology of geometric visualization"
- Reply: Herman Jurjus: "Re: psychology of geometric visualization"
- Messages sorted by: [ date ] [ thread ]
Date: Wed, 07 Jul 2004 01:48:18 -0700
In article <qQLGc.39247$WB5.2599@pd7tw2no>, Elaine Jackson
<elainejackson7355@home.com> wrote:
> Visualizing things in three dimensions doesn't come naturally to me, so I'm
> always trying to improve my skills in that area. If anybody knows any
> references
> or has any suggestions to make, I'd be glad to hear about it. In case that's
> too
> general for you, here's the specific thing that's on my mind at the moment: a
> cube rotating on one of its diameters. Apparently many (most?) people consider
> it obvious that the cube coincides with itself every 120 degrees, but for me
> it's extremely difficult to "see" that. In fact, I went to the trouble of
> writing down an actual (analytic) proof. But of course that misses the point.
> What I'd really like to know is this: if this conclusion seems completely
> clear
> to you, can you give me any insight into how you arrive at it?
>
I've always thought "visualization" is a misleading term. I prefer
"spatialization". It's not so much seeing as understanding the spatial
relations of objects to each other, perhaps even in a kinesthetic
fashion.
In terms of your specific question, it helps to be able to have a Kodak
view of a cube and see it rotate, but that's not really necessary. The
idea is to understand the spatial relations. The first key thing in
any spatialization is to put yourself in the proper
position/perspective. This can often be what makes or breaks a good
spatialization. In this case, what this means is to pick the right
angle to view the cube.
My preferred view (at the moment anyway) is to pick the cube so that
the axis of rotation (in this case, a long diagonal of the cube) is
pointing straight between my eyes and is perpendicular to the axis of
my body. The cube is small, about the size of my head and floating a
foot and a half in front of my head.
The second step is to now pick the relevant spatial features from this
position. Of course, in picking the right position you had to have an
idea of this. Notice that from this view, you can only see three sides
of the cube. The sides that touch the corner that's part of the axis
of rotation and visible to you. [The cube is opaque by the way]
The key thing here is to note that there's a 3-fold symmetry. All of
those 3 sides look the same to you if you spin the cube on the axis.
It's clear that a 120 degree rotation will preserve the front of the
cube. But you may be worried about the back. But the point here is
that if you understand how the three back sides fit together, you
realize that the back looks the same as the front, except by a 180
degree rotation. So when you rotate by 120 degrees, the back is
preserved also.
You can think of looking at the back and front separately from the same
kind of position as I've described. But I like to think of it as I'm
feeling the back side with my hands. Once you understand the front
side and you remember how a cube feels in your hands, you can just
slide your hands from the front of the cube to the back and just feel
how it is.
[All through this I did have a weird Kodak picture view of the cube and
for some reason it was purple. I have no idea why.]
- Next message: Eric Gisse: "Re: A little lesson for sqrt(144) year olds."
- Previous message: Michael N. Christoff: "Re: Topology and Dynamical Systems"
- In reply to: Elaine Jackson: "psychology of geometric visualization"
- Next in thread: Herman Jurjus: "Re: psychology of geometric visualization"
- Reply: Herman Jurjus: "Re: psychology of geometric visualization"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
