1
$\begingroup$

I've been given an assignment to create a shape that has a Beta symmetry of 135°.

My textbook defines Beta symmetry as:

Beta is the rotational symmetry of a part about its axis of insertion. The magnitude of rotational symmetry is the smallest angle through which the part can be rotated and repeat its orientation. For a cylinder inserted into a circular hole, beta equals zero.

There are several shapes that have 135° rotational symmetry - octagon, 8 pointed star, for example, but those both also have 45° rotational symmetry, and since Beta symmetry is the "smallest angle", does a shape even exist that has Beta symmetry of 135°? This is driving me crazy!

  • 1
    You can have a magnitude of symmetry at a specific angle only if that angle divides 360 degrees. – 2017-02-17
  • 0
    Your reasoning sounds correct to me, although I have never heard the term beta symmetry before: is it from industrial design? Could there be something with moving parts in such a way that turning it $3\times 135$ degrees (a full turn plus 45) is different from turning it $45$ degrees? That would be one way to escape the standard argument that any shape with 135 degree symmetry also has 45 degree symmetry. – 2017-02-17
  • 0
    @Paul Probably I'm misunderstanding what you mean, but I would think what you've written implies that regular $7$-gons (among others) can't exist, since $7$ doesn't divide $360$. – 2017-02-18
  • 0
    @pjs36, no, it's the angle measure that has to divide 360, not the number of sides. – 2017-02-18

1 Answers 1

0

Alpha symmetry is the largest angle a part would have to be turned about an axis perpendicular to the insertion axis.

Beta Symmetry is the largest angle the part would have to be rotated about the insertion axis for mating.

Alpha and Beta symmetry actually range from 0 to 360° (instead of the intuitive 0 to 180°) because it is assumed that the worst case rotation is used. reference Design.