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# The Limits of Flatness

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It's just generalizing the concept of "slope of a line". How many slopes does a curve have?

posted by muddgirl at 4:00 PM on July 19, 2012 [1 favorite]

posted by IAmBroom at 10:31 AM on July 20, 2012

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# The Limits of Flatness

July 19, 2012 1:27 PM Subscribe

Are there an infinite number of 2D shapes?

I'm pretty sure--but at times not--and want to double-check. If you have the logic/science behind your answer, that'd also be cool.

I'm pretty sure--but at times not--and want to double-check. If you have the logic/science behind your answer, that'd also be cool.

Yes. Even just the set of "shapes of n sides with equal length sides" is infinite.

posted by tyllwin at 1:32 PM on July 19, 2012

posted by tyllwin at 1:32 PM on July 19, 2012

Yes. Consider the shapes with n equal sides, where n is any integer. This already gives you as many unique shapes as there are integers, in other words, an infinite number.

posted by Pre-Taped Call In Show at 1:33 PM on July 19, 2012

posted by Pre-Taped Call In Show at 1:33 PM on July 19, 2012

If a shape is simply the lines that connect the points on the corners...

And there are an infinite number of possible x,y coordinates for those corners...

Then there are an infinite number of shapes.

posted by DWRoelands at 1:33 PM on July 19, 2012

And there are an infinite number of possible x,y coordinates for those corners...

Then there are an infinite number of shapes.

posted by DWRoelands at 1:33 PM on July 19, 2012

Yes, there are an infinite number of 2d shapes. As proof by induction, take an isosceles triangle with a base of length 1. This triangle can have any real height between 0 and infinity, hence an infinite number of isosceles triangles.

posted by justkevin at 1:35 PM on July 19, 2012

posted by justkevin at 1:35 PM on July 19, 2012

To answer the question: Yes. 2d, 3d, 4d, they all have infinite shapes. The # of dimensions are irrelevant.

Infinity has lost a lot of its luster over the years.

posted by Sphinx at 1:46 PM on July 19, 2012 [1 favorite]

Infinity has lost a lot of its luster over the years.

posted by Sphinx at 1:46 PM on July 19, 2012 [1 favorite]

I do not mean to troll here, but as stated before, restless_nomad's answer makes no sense at all.

In general, the concept of a side, as in sides for polygons, does not apply to circles.

Also, it depends very much on the theory at hand what terms like "infinity" and "infinity - 1" denote. In all cases, they are not numbers. If at all meaningful, infinity - 1 would be regarded as infinity. Thus, to say that a property is satisfied for infinity (whatever that means), but not for infinity - 1 makes no sense.

Lastly, the conclusion does not follow from the previous statements at all.

posted by arhammer at 2:52 PM on July 19, 2012 [3 favorites]

In general, the concept of a side, as in sides for polygons, does not apply to circles.

Also, it depends very much on the theory at hand what terms like "infinity" and "infinity - 1" denote. In all cases, they are not numbers. If at all meaningful, infinity - 1 would be regarded as infinity. Thus, to say that a property is satisfied for infinity (whatever that means), but not for infinity - 1 makes no sense.

Lastly, the conclusion does not follow from the previous statements at all.

posted by arhammer at 2:52 PM on July 19, 2012 [3 favorites]

There are not only an infinite number of 2D shapes, there are an infinite number of different triangles as well. There are an infinite number of numbers between 1 and 179 (1.01, 1.001, 1.0001, etc.) which are all angles that triangle can have at one of its points. There are an infinite number of regular 2D shapes, like justkevin said, and each one of these shapes can then be altered like the triangle.

Infinities of infinity.

Infinity is weird like that.

posted by Hactar at 3:03 PM on July 19, 2012

Infinities of infinity.

Infinity is weird like that.

posted by Hactar at 3:03 PM on July 19, 2012

And thus Hactar has proved that it's aleph one, not just aleph zero.

posted by Chocolate Pickle at 3:49 PM on July 19, 2012 [1 favorite]

posted by Chocolate Pickle at 3:49 PM on July 19, 2012 [1 favorite]

*In general, the concept of a side, as in sides for polygons, does not apply to circles.*

It's just generalizing the concept of "slope of a line". How many slopes does a curve have?

posted by muddgirl at 4:00 PM on July 19, 2012 [1 favorite]

Surprisingly, infinities come in assorted sizes. Aleph-0 is the cardinality of integers; a bigger number called

There's a cool proof illustrating how you can have a number bigger than aleph-0.

Since you can represent (at least some) geometric curves parametrically with real-valued parameters, the number of geometric curves is at least

posted by naturetron at 4:07 PM on July 19, 2012 [1 favorite]

*c*is the cardinality of the real numbers. (Aleph-1 might be the same thing as*c*, but it's undecideable.)There's a cool proof illustrating how you can have a number bigger than aleph-0.

Since you can represent (at least some) geometric curves parametrically with real-valued parameters, the number of geometric curves is at least

*c*.posted by naturetron at 4:07 PM on July 19, 2012 [1 favorite]

Imagine a rectangle whose length is twice its width.

Now imagine a rectangle whose length is three times its width.

Now imagine a rectangle whose length is four times its width...

Even between the first two there are an infinite number of shapes:

Imagine a rectangle whose length is 2.10457 times its width.

Imagine taking large sheets of paper and drawing closed doodles.

Do you think you'd run out of shapes?

posted by weapons-grade pandemonium at 7:02 PM on July 19, 2012

Now imagine a rectangle whose length is three times its width.

Now imagine a rectangle whose length is four times its width...

Even between the first two there are an infinite number of shapes:

Imagine a rectangle whose length is 2.10457 times its width.

Imagine taking large sheets of paper and drawing closed doodles.

Do you think you'd run out of shapes?

posted by weapons-grade pandemonium at 7:02 PM on July 19, 2012

I also wanted to mention fractals. Look 'em up, you'll like 'em.

posted by windykites at 9:12 PM on July 19, 2012

posted by windykites at 9:12 PM on July 19, 2012

As naturetron pointed out, there are different kinds of infinities. For example, there's the integers { ... -3, -2, -1, 0, 1, 2, 3...} which are countably infinite and then there's the reals, which are uncountably infinite.

Since you can draw justkevin's isosceles triangle with any real number greater than 0 as the height, the number of 2D shapes is uncountably infinite, which is bigger in some sense than being merely countably infinite.

I think you can use restless_nomad's idea to get a proof that the set of 2D shapes is at least countably infinite but it turns out it's significantly larger than that.

posted by Aizkolari at 8:31 AM on July 20, 2012

Since you can draw justkevin's isosceles triangle with any real number greater than 0 as the height, the number of 2D shapes is uncountably infinite, which is bigger in some sense than being merely countably infinite.

I think you can use restless_nomad's idea to get a proof that the set of 2D shapes is at least countably infinite but it turns out it's significantly larger than that.

posted by Aizkolari at 8:31 AM on July 20, 2012

restless_nomad: If a shape has an infinite number of sides, it's a circle.Julia sniffs at being called a "circle", and turns her backs on you.

posted by IAmBroom at 10:31 AM on July 20, 2012

This thread is closed to new comments.

posted by restless_nomad at 1:29 PM on July 19, 2012 [9 favorites]