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wiki>Mats Halldin: /* Cube with interior cube */ +image

2011-04-24T09:37:06Z

‎Cube with interior cube: +image

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Introduction to topology.

= Euler's theorem =
; [http://en.wikipedia.org/wiki/Euler%27s_theorem Euler's theorem]: Let P be a polyhedron which satisfies:
: (a) Any two vertices of P can be connected by a chain of edges (i.e. there are no faces missing and no "Non-Manifolds".)
: (b) Any loop on P which is made up of straight line segments separates P into two pieces (i.e. a mesh without holes like in a donut.)
: Then <math>v-e+f=2</math> for P.

== Default cube ==
[[File:Theory-Euler-DefaultCube.png|thumb|Default cube selected in edit mode]]
If we add a cube and tab into edit mode, all vertices are select by default and above the 3D view Blender is informing us that
: <code>Ve:8-8 | Ed:12-12 | Fa:6-6 | Cube</code>
which, of course, means we have selected all eight vertices, twelve edges, and six faces of our object <code>Cube</code>.

According to Euler's theorem
: <math>v-e+f=2</math>
if we take the number of vertices (''v''), subtract the number of edges (''e''), and add the number of faces (''f'')
: <math>8-12+6</math>
we get 2.
{{Clr}}

{| align="center"
| [[File:Theory-Euler-DefaultCube-FaceExtruded.png|thumb|A face extruded]]
| [[File:Theory-Euler-DefaultCube-FaceExtruded-triangulated.png|thumb|Triangulated]]
|}
If we extrude a face from our cube, Euler still returns
: <math>12-20+10=2</math>.
And if we hit {{Shortcut|Ctrl|T}} to transform all quads into tris, Euler still returns
: <math>12-30+20=2</math>.
{{Clr}}

[[File:Theory-Euler-DefaultCube-FaceDeleted.png|thumb|Face deleted]]
On the other hand, if we remove a single face from our original cube, Euler returns
: <math>8-12+5=1</math>
instead.
{{Clr}}

=== Other simple primitives ===
Let's test Euler's theorem on other well-known geometric primitives:

Now, add a default '''UV Sphere''' and tab into edit mode. Blender now informs us that
: <code>Ve:482-482 | Ed:992-992 | Fa:512-512 | Sphere</code>
and Euler returns
: <math>482-992+512=2</math>.
Extruding (<math>486-1000+516=2</math>) or deleting (<math>482-992+511=1</math>) a face from the UV sphere is not changing the output from Euler's formula.

Same goes for the default cylinder
: <code>Ve:576-576 | Ed:1152-1152 | Fa:576-576 | Cylinder</code>
: <math>66-160+96=2</math>.

We might first hope that this is a golden rule that applies unconditionally to all meshes.

== Some more complex cases ==
The default torus
: <code>Ve:576-576 | Ed:1152-1152 | Fa:576-576 | Torus</code>
: <math>576-1152+576=0</math>.

=== A monkey ===
Cubes, spheres, and even toruses are well-known geometrical shapes, but what about {{Literal|Suzanne}}, our monkey? In edit mode, Blender tell us
: <code>Ve:507-507 | Ed:1005-1005 | Fa:500-500 | Monkey</code>
and Euler returns
: <math>507-1005+500=2</math>.
Looks familiar.
However, if we deselect everything and {{Shortcut|Ctrl|Shift|Tab|M}} to select {{Literal|Non-Manifolds}}, Blender selects two pairs of loops, namely the holes for the eyes. Apparently, the monkey is topologically distinct from the default cube.

Furthermore, if we {{Shortcut|Ctrl|T}} on the monkey and select all tris, we get
: <math>507-1472+968=3</math>.
Selecting non-manifolds again returns an additional edge near the nose.

Uh-huh, topology-wise something is wrong here.

=== Cube with interior cube ===
[[File:Theory-Euler-DefaultCube-InteriorCube.png|thumb|Interior cube]]
Select all vertices of the default cube, {{Shortcut|Shift|D}} to duplicate, and {{Shortcut|S}} to scale it down. Our mesh now has two cube, one inside the other. Blender tells us
: <code>Ve:16-16 | Ed:24-24 | Fa:12-12 | Cube</code>
and Euler returns
: <math>16-24+12=4</math>.

This is new...
{{Clr}}

=== Cube with interior faces ===
[[File:Theory-Euler-DefaultCube-InteriorFaces.png|thumb|Interior faces]]
: <math>27-54+36=9</math>
{{Clr}}

=== Möbius strip and Klein bottle ===
A Möbius strip
: <math>32-48+16=0</math>

A [http://mathworld.wolfram.com/KleinBottle.html Klein bottle] is what you get if you glue two Möbius strips together along a common edge.
: <math>624-1248+624=0</math>

Odd as they come, these two meshes are, according to Euler's theorem, similar to a circle...

== Conclusions ==
Is there any logic to this?
* Circle=0
* Disc=1
* Cylinder=2

* ''If two polygons meet they do so in a common edge, and each edge of a polygon lies in precisely one other polygon.''
* At any vertex, faces fit together to form a piece of surface around that vertex.
* Is there a loop that does not separate the mesh into two distinct parts?

= References =
* Armstrong, M. A. (1983), ''Basic Topology'', Springer, pp 1–12, ISBN 0387-90839-0.

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wiki>Mats Halldin: /* Cube with interior cube */ +image

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