Alex Karassev

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Why do we need more than three dimensions?

• Alex Karassev

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Disclaimer
We (almost) wont talk about
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But
We will talk about
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Some boring stuff
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Some boring stuff
x1
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Similarly
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The main question is
What do we need it for?
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Configuration space
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Canadian-built space stationrobotic arm
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Linkage with two segments
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Linkage with two legs
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Torus
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Linkage with two legs
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Example
Toroidal spiral
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More configuration spaces

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How about three legs?
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So, what do we get?
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3-dim torus in action
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Linked circles and knots
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Strange 4D
In the four-dimensional space
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Or is it?
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Now we try to use 4th dimension
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How do we place 3-dim torus in 4-dim space?
Space
Time
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What was that about the orange?
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What was that about the orange?
Cut with R3
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Image recognition
Leonardo da VinciLady with an Ermine"
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Yes!
n 1,000,.,000
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Example consider 2 points such thateach point
can be either green or red
0
1
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How to tell the difference between A and B ?
A A A A A B B B B B
A
A
B
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Cubes
(1,1,1)
(1,1,0)
(0,0,1)
(1,0,1)
(1,1)
(0,1)
(0,1,1)
0
1
(0,0)
(1,0)
(0,0,0)
(0,1,0)
1-dim
2-dim
3-dim
For all k we have 0 xk 1
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Hypercubes
Exercise
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Projections
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4-cube in action
Rotations about planes!
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Rotation about a plane
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Rotation about two planes
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What was that picturein the background?
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Another puzzle picture
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Thank you!