Canceling Cones

I don’t think the reconciling of the non-smoothable cone points in that last post was terribly plain to see. Perhaps if you could manipulate the models yourself, it would be more apparent.

These two pictures show the essence of the move. Take a curly Q crossed with an interval. Shrink the middle loop down to a point. Then pull apart the resulting cones. Reverse to smooth a pair of cones.

Picture 1

Here’s the same guys bent a bit.

Picture 2

Continue on to see how this applies to that octaheralishy Boy’s surface.

A neighborhood of an edge connecting two cone points has been colored green.

Picture 3 Picture 4

Now we’ll split it apart from the rest.

Picture 5 Picture 6
Picture 7 Picture 8

Finally, let’s bend this thing a bit to see how it fits in with the very first pictures.

Picture 10

Picture 11

Now you can do the cone canceling. Do this to two more edges of the octaheralishy Boy’s surface to get a smooth immersion.

~ by Ken Baker on June 18, 2008.

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