Here are some vague ideas I would like explore in a workshop at Mathcamp 2010. They are untried, and I hope we can refine them and come up with more ideas.

Google for "Flash Mob" to find interesting examples of public group activities where many people read a pre-designed plan that is promoted through a web site, then come together at a specific time to follow the plan and do something interesting, and then disperse. See, for example:

http://en.wikipedia.org/wiki/Flash_mob

http://en.wikipedia.org/wiki/Improv_Everywhere

I would like to design and try out some math-oriented flash mob activities, which I'll call "math-mobs." Below are some ideas I have in mind. My plan for Mathcamp is that we first have a session in which we develop these ideas further, come up with new ideas, chose some, and make specific instructions of what participants should do. Then we announce to the rest of the camp a time when we'll try them out. Probably the try-out time would be the next day or at least several hours later, so we can advertise to the whole camp and more people can participate.

1. Giant Mobius strip. (Worlds largest?) Everyone brings a bed sheet with them to a specified outdoor location. We stand in a big circle, a sheet-length apart from each other. We hold the sheets so they connect in a giant cylinder. Each person holds the top corners of their sheet and the bottom corners of the next sheet. But we have a twist in it as you go around, so it is a giant Mobius strip. Probably the sheets would be folded in half the long way, so they are not wider than a person's armspan. Next we do a slow gentle rotation, a kind of "wave" that goes around the circle.

2. Giant Complete Graph. We have n people come together and make K_n using some kind of ribbon they bring with them. Perhaps each person gets a roll of the flourescent surveyor's plastic "flagging tape" (http://www.bestmaterials.com/detail.aspx?ID=8923). We stand in a circle an arms length apart. Each person ties the end of their roll of tape to their left wrist, then passes the roll to the person 1 to their right. When you receive the roll from the person on your left, you wrap it once around your left wrist and pass it 2 people to your right. Wrap it again, then pass it 3 people to the right... After it goes n/2 people to the right, the view from above will be a complete graph as in these "string art" images:

http://en.wikipedia.org/wiki/Complete_graph

3. Human exponential tree. One person starts and extends his/her two arms out in front. Two people then come and stand so an extended hand is at their back. They each extend their arms. Four people come and become the next level. Then eight, etc. How far can this go on? At some point the next level can not all fit, illustrating the practical limits of exponential growth.

4. Square-root-of-five squares construction. Pick a place with a square-grid tiled floor. Everyone brings one (or several?) of five different things. (We specify five fruits?) One at a time everyone adds an element to the growing pattern, one in each floor tile to make the "knight's move" pattern that divides the grid into five subgrids, each with square of five times the area, i.e., sqrt(5) times the edge length. We leave it on display for some minutes, then each person takes away as many items as they brought.

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D E A B C D E A B C D E A B C

B C D E A B C D E A B C D E A

E A B C D E A B C D E A B C D

C D E A B C D E A B C D E A B

A B C D E A B C D E A B C D E

D E A B C D E A B C D E A B C

B C D E A B C D E A B C D E A

E A B C D E A B C D E A B C D

C D E A B C D E A B C D E A B

D E A B C D E A B C D E A B C

B C D E A B C D E A B C D E A

E A B C D E A B C D E A B C D

C D E A B C D E A B C D E A B

A B C D E A B C D E A B C D E

D E A B C D E A B C D E A B C

B C D E A B C D E A B C D E A

E A B C D E A B C D E A B C D

C D E A B C D E A B C D E A B

5. Giant Tessellation. Everyone brings a handful of pencils and/or pens. We don't write with them; they are just convenient "sticks" to build with. Somehow we manage to lay them out on a floor somewhere as the edges of an interesting tessellation. We look at it for some minutes, then each person takes away as many items as they brought. What would be a good tessellation for this? How would we make it accurately enough given writing implements of varying lengths?