Möbius strip
A Möbius strip is a surface with only one side and only one boundary. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface.
Know more:
http://mathworld.wolfram.com/MoebiusStrip.html
Animation via reddit
#maths #mobiusstrip #animation
The motion is extremely cool... and it's a Menger fractal to boot. I'm surprised I haven't thought of experimenting with a Mobius strip yet - they aren't much more complicated than a torus to construct.
ReplyDeleteCan you 3D print one though...
ReplyDeleteIt's a real form Sam Collett, if that is what you're getting at. It's not an Escher-like impossible form.
ReplyDeleteIt does look unreal, does anything in nature take that form though?
ReplyDeleteKlein bottle looks interesting too
I don't believe it occurs naturally. I meant it isn't a logically impossible form. Just a funky-interesting one.
ReplyDeleteFor extra credit, make your mobius strip out of fairly wide paper, and then cut it in half lengthwise. Then cut that in half again and so forth. The results are kind of weird and worth exploring.
ReplyDelete+Sean Walker, the Mobius band is embedded in the space of rotations of 3-dimensional space. In fact, in that space of rotations there is a thing that is called "the projective plane." It is the union of a Mobius band and a disk where the two are joined along their boundaries.
ReplyDeleteA rotation of 180 degrees clockwise about a positive vertical axis is the same thing as a rotation 180 degrees counterclockwise about a negative vertical axis. If you tilt the axis, say, along the prime meridian/ international dateline, from the equator near the southern African coast (South of Guana) the antipodal axis will tilt towards Malaysia (rough geography). You can find a slim band around the longitude 0 and 180 where the rotations form a Mobius band.
A strip mobius has two edges. But the 3D sculptural version only has one edge.
ReplyDeleteBob Calder according to the mathematical definition of a "closed" Mobius band it has one bounding circle; it is a 2-dimensional surface that topologocally identical to one that is formed from the set of all points (x,y) in the plane with 0 less than or equal to x less than or equal to 10 (use \le for that inequality) and
ReplyDelete-1 \le y 1. The points (0,y) are identified with (10,-y). In this way, the short ends under-go a 1/2 twist as they are being "glued" together.
Scott Carter I tried to work through what you had described and was gapping a bit. At the highest level, I believe your point is that Mobius strips manifest 'naturally'. Correct? Digging in slightly deeper, I believe your point was that this natural manifestation can be arrived with a the space of 3D rotations. However, the mechanics of arriving at the thin strip you mentioned was lost on me. I think a more mathematical description would help - perhaps based on a spherical coordinate system. I believe your point is that when you rotate in some manner, some subset of points on the surface of a sphere move effectively along a path of a Mobius strip. Correct?
ReplyDeleteSean Walker In order to understand the Mobius band as it is embedded in the space of rotations of 3-dimensional space, it takes a little time to first describe that space. Any rotation of 3-space can be restricted to a soccer ball centered at the origin. This is independent of any FIFA scandals. The ball will rotate along some axis and through some angle. The angle is any angle from -180 to +180 degrees.
ReplyDeleteWe can identify the set of these rotations with points in the inside of a (different) ball of radius 1. The axis determines a direction, and the angle determines a ratio along that axis. So if you rotate 20 degrees about the vertical axis, the start at the center of the ball and move vertically a distance 1/9 along that axis. If you move 45 degrees, then move a quarter of the way up the axis. To each rotation, we have a point in the ball. If the rotation is negative, then move down.
Now a positive 180 rotation about the vertical, is the same as a negative 180 degree rotation about the vertical. So the points on the boundary of the ball are identified antipodally.
Think of a vertical band inside this different ball, again centered along the axis. For definiteness, let's suppose that the band is aligned with the prime meridian and the international dateline. and it extends to a latitude of 80 degrees north and south. The rotations that correspond to that band form a Mobius band in then space of rotations.
I hope this helps. If not, look up SO(3) in Wikipedia.
Scott Carter I didn't study topology so thank you for the second pass on that. I understood your second description better than your first (Wikipedia on this is not very helpful for me as it covers too much material with very limited exposition).
ReplyDeleteIn any case, I understand now that you were not intending to provide an example of a natural occurrence of a Mobius in nature. A band in SO(3) undergoing a composition of rotational transformations is an alternate way of conceptualizing or imagining a Mobius as opposed to a natural occurrence of one. Otherwise any mathematical relationship or observation effectively would represent a natural occurrence and that would definitely be stretching the sense of the term.
Sean Walker , well yeah. I think of the space of all rotations of this 3-d space that we imagine ourselves living in as a natural space. It is the set of transformations of 3-d space just as a circle is the set of rotations of the plane. There are many different Mobius bands therein. They all have roughly the same description. Corina Marinescu 's posts span many sciences and the medical arts. They are consistently interesting!
ReplyDeleteDanke Scott :)
ReplyDelete