# Alternative revolutions

Mon 3 October 2022
Tagged: puzzle

You can use the profile of an object that has rotational symmetry, but revolve it around a different axis, to produce a different object that has the same profile. It feels like there's a puzzle idea in there somewhere, but I can't work out the best way to turn it into an actual puzzle, so if you have any ideas please let me know.

I suspect the kind of people who might be good at this puzzle are those who do a lot of CAD and those who do a lot of lathe work, because they both involve grappling with the equivalence between the 2d profile and the 3d revolved shape.

## Profiles

Here's a rough profile of a chess pawn, and the shape you get by revolving it around the tall vertical edge:

The idea behind "alternative revolutions" is that we can revolve the same profile around a different edge to get a different object:

Hopefully you see that this has the same profile, but is a completely different shape.

We also have the option to make negatives by taking a negative of the profile:

This is the same profile, except now the right hand side, instead of the left hand side, is "inside" the shape.

And we can do the same with the horizontal axis instead of the vertical axis. For some reason the 3D shape created by this profile is very confusing to look at, but it should make sense if you keep remembering the profile that it's made out of.

(This part would exactly mate with the first "alternative revolution" above).

And the same thing with the other horizontal edge:

The mating part for which is:

We can sweep around an axis that is spaced away from our profile, to leave a hole in the middle:

And our profile can either face outwards (above) or inwards:

And the same with the horizontal rotation axis:

And of course the same is also possible with the negative version of the profile, which among other things lets us make a mating part for the standard chess pawn:

We also have the option to tilt the rotation axis if we want to:

This lets us smoothly interpolate between the version with the vertical rotation axis and the one with the horizontal rotation axis:

With a bit more creative license, we could decide that we don't want the hollow cones that give the game away, and we could easily cap off the top and bottom with flat surfaces.

(Sadly I can't think of a way to smoothly interpolate between the positive and negative version of the profile.)

So there are quite a lot of different shapes we can make, given just one profile, by having the option to turn the profile negative, and varying the position and direction of the rotation axis.

## Puzzle ideas

Identify common objects: We give the player a bunch of objects that are made by revolving common object profiles (some positive, some negative) in uncommon ways. The player has to figure out what common object each shape is based on, and then... what? Concatenate the words to form a password? Just gain the satisfaction from having solved it?

Matching pairs: We give the player a bunch of objects that are made out of contrived profiles (i.e. arbitrary shapes we make up, not necessarily related to objects that the player would know). The player has to work out which of the objects share profiles and pair them up. And then... there's a number on some of the objects and a letter on the others, and they have to put the letters together in order of the numbers to make a password?

Odd one out: We give the player a bunch of objects that can be made out of contrived profiles or common ones, and they have to locate the only object that doesn't share a profile with any of the other objects (or, along similar lines, they have to locate the only pair of objects that do share a profile).

Data encoding: We can come up with a way to encode a small amount of data in the profile (as an existence proof: make a large radius for a 1 bit and a small radius for a 0 bit, but more interesting stuff is possible) and the player gets the objects and has to decode the data we've given them. Maybe in combination with "odd one out" or "matching pairs" to work out which pieces of data are actually useful and in what order?

Normally I prefer (dis-)assembly puzzles, or something that acts like a box and you have to solve the puzzle to retrieve an object, but I can't see a way to readily adapt the "alternative revolutions" to anything interesting. Probably the best option is to just use it as a matching puzzle to encode a combination for a padlock in an escape room.

What better ideas do you have?

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