…literally.

In his

*Meditations on First Philosophy*, Descartes distinguished between imagination and conception, or between mental images and concepts. Thus he supposed that one can conceive of a chiliagon, a polygon of a thousand sides, although one cannot form a mental image that represents it—none, at any rate, that wouldn’t represent a circle equally well. This shows that imagination has its limits, and that one can conceive of things that one cannot (adequately) imagine.In order to discern what these limits are, I invite my readers to test the limits of their own imaginations in an experiment based on Descartes’ example. Since it concerns only what you can imagine, you don’t have to resort to a lab—in this case a “thought experiment” and a real experiment coincide!

Now, I’m sure you can imagine a polygon with the least possible number of sides—a triangle. I’d wager that you can also imagine a square, a pentagon, a hexagon… but not a chiliagon. It would thus seem that there is some number

*n*such that you can imagine an*n*-sided polygon but not an*n*+1-sided polygon. (Note: For the purposes of this experiment you only count as imagining a polygon if you can imagine the whole thing at once.) Let’s call this*n*-sided polygon your*limit polygon*. I have two questions: First, what is the number of sides of your limit polygon? Second, do you notice anything about the phenomenology of your limit polygon? If so, what?
## 7 comments:

I can't get larger than eight without it become a bit unclear how many it has; and mostly the octagon itself is because I can imagine a stop sign, and thus because the octagon is the most-sided polygon I regularly see already.

I can get to 32, not by imagining each consecutive polygon, but by starting with 8 and imagining the doubling of each side twice. But that's actually quite difficult, and its easier to go back to 16.

So maybe there are different strategies that can be utilized in order to imagine polygons with higher sides?

"Insert" a new side at each of the vertices of an octogon to go from 8 to 16...I have just enough imagination to pull that off with an all-at-once estimate. It's "fuzzy" but I think with practice I could do it. I strongly suspect that N will always be even because we'll use a doubling algorithm like the one I describe. I also suspect we can only double once beyond the largest polygon in our visual experience--for most, an octogon.

To go to a new nerd level, go 3D and base your image off of the polyhedrons used in RPGs like Dungeons and Dragons. The largest standard die is a 20-sided, which I can form a pretty solid mental image of on the basis of experience. I find I have no way to add an imaginary side to this image however. The 100-sided die (quite a chimera) is something I do not use enough to form an image of but I'm not sure I could even if I did use it often. It's practically a sphere.

Finally, just to resoundingly agree with Descartes, the staggering ease of conceptualizing these additions, compared to the intense labor of imagining them, is highly suggestive.

Cool post!

This is weird, but I'm being straight with you here: I have no problems imagining polygons with 3, 4, 5, 6, and 8 sides. But I'm having trouble with 7, 9, and anything greater than 9.

Given my own difficulties here, I'm having trouble taking seriously the various claims to imagine, say, 32-sided polygons. I've got to wonder if these people correctly grasp Descartes' important distinction. Are they sure they can distinguish, in *imagination* as opposed to thought, 30-sided pgons from 28-sided and 32-sided?

It might be useful here to have data on what sorts of limits there are of perceptual discriminations of real polygons people can make, in both simultaneous and serial presentations of polygon pairs.

Also relevant: Can people tell polygon side number *just* by looking, that is, without counting? I'm skeptical that they can do this when n gets into the double digits. Anyway, this is important because counting seems to be a nonimagistic, thought process and thus on the wrong side of Descartes' distinction.

I've got a mental image of 24 sides. I started with a Star of David (12 sides) and added little triangles in the corners. It's a very clear picture in my mind.

A typical 5 pointed star has 10 sides.

A W in common calligraphy has 13, 16 or sometimes 24 sides. I was a student at UW, and the W on our home page has 21 sides. K has 11 sides.

Sorry, I must have been trying too hard. A swastika has 20 sides.

Using these shapes I can imagine polygons of 3,4,5,6,7,8,9,10,11,12,13,14,16,20,21 and 24 sides.

Starting in a square rounding the corners to 20 then 25 each got me to 100 sides. I'm sure I could get more by expanding something with more then 4 sides in this fashion.

-RaOb

When doing this thought experiment, I used a triangle first. Then every time I blinked, I added a number of random sides. The number got to a chiliagon pretty fast, then it doubled. By this time I was zooming in and out to see the effect it had. It was getting to be a perfect circle, then I folded the sides in and got a snow flake shape. I added more sides and got a version of the Mandelbrot bug's leg. Then added even more sides, this is when the picture went over an imaginary horizon. I quickly shrunk it down. By this time I could imagine the universe folding into a 2-d shape by the googleplex^googleplex sides, and that's when I stopped this experiment because though it was fun, infinity gives me a headache.

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