How to build an n-dimensional object?

Ken Arroyo Ohori

ABE010: Capita Selecta 20.11.2014

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The 24-cella “simple” 4D object

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The 24-cell

24 0D vertices 96 1D edges 96 2D faces

24 3D volumes 1 4D hypervolume

a “simple” 4D object

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objects in more than 3D are complex!

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Some background

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0D: a vertex

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0D: a vertex

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1D: an edge

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1D: an edge

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1D: an edge

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1D: an edge

a 1D object can be described by its 0D boundaries

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2D: a face

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2D: a face

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2D: a face

a 2D object can be described

by its 1D boundary

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2D: a face

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2D: a face

which can be described by its 0D boundaries

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2D: a face

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3D: a volume

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3D: a volume

a 3D object can be

described by its 2D boundary

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3D: a volume

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3D: a volume

which can be described by

its 1D boundaries

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3D: a volume

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3D: a volume

which can be described by

their 0D boundaries

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4D, 5D, …

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= 8 cubes

however, there is a problem in practice…

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2D: a face

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2D: a face

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2D: a face

a 2D object is described by a set of 1D

objects

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2D: a face

a “soup” of line segments

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2D: a face

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2D: a face

same coordinates

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2D: a face

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nextnext

nextnext

next

next

2D: a face

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nextnext

nextnext

next

next

build an object!

2D: a face

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3D: a volume

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3D: a volume

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a “soup” of faces

3D: a volume

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3D: a volume

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build a volume

from a set of faces

The solution: incremental construction

• Start from a set of 0D vertices

• Connect them to form 1D edges

• Connect these to form 2D faces

• Connect these to form 3D volumes

• …

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Building two tetrahedra

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Building two tetrahedra(a) 1

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3

4

5

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Building two tetrahedra(b)

a

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2

5

4

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Building two tetrahedra

b

(c)

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2

5

4

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Building two tetrahedra(d)

d

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2

5

4

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Building two tetrahedra(e)

c

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2

5

4

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Building two tetrahedra

e

(f)

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2

5

4

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Building two tetrahedra(g)

f

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2

5

4

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Building two tetrahedra

g

(h)

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Building two tetrahedra

c

a

ge

fd

b

(a)

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Building two tetrahedra

c

a

ge

f

b

d

(b)

done!

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Build a tesseract

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Build a tesseract

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build each cube

separately

Build a tesseract

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build each cube

separately,

then join them

Build a tesseract

done!

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Methodology• Analyse the problem

• Split it into small, manageable subproblems

• Try sketches/ideas on paper

• When mature enough, build a program to test these

• Start with simple shapes, move towards more complex ones

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Thank you!Ken Arroyo Ohori

g.a.k.arroyoohori@tudelft.nl tudelft.nl/kenohori

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Images from:

• http://commons.wikimedia.org/wiki/File:Stereographic_polytope_24cell_faces.png

• http://blogs.lt.vt.edu/foundationdesignlab/category/materials/

• http://commons.wikimedia.org/wiki/File:Schlegel_wireframe_8-cell.png

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More info

Ken Arroyo Ohori, Guillaume Damiand and Hugo Ledoux. Constructing an n-dimensional cell complex from a soup of (n-1)-dimensional faces. In Prosenjit Gupta and Christos Zaroliagis (eds.), Applied Algorithms, Volume 8321 of Lecture Notes in Computer Science, Springer International Publishing Switzerland, January 2014, pp. 37–48.

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