T's in the plane

You can fit uncountably many lines in the plane (e.g. by having them parallel). You can also fit uncountably many circles in the plane (e.g. by arranging them concentrically).

Can you fit uncountably many capital letter "T"s in the plane? Give an example or prove that you can't.

#Geometry

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

For every length L, we can think of the plane divided up in to square cells of side length L. Since there are countablly many cells in the grid, as least one cell must contain parts of uncountably many Ts. But there is no dense way to stack Ts, unlike lines and circles. I don't have a quick geometric argument as to why not, but the greatest number of parts of distinct Ts I can get arbitrarily close to each other is 10---that's a lot, but it's countable

Mark C - 5 years, 1 month ago

I think that's a really good start (and intuitively the whole answer). Try taking little boxes that contain the intersection point (like where the two lines cross) of each T instead of boxes around whole Ts. If you can argue that you only need countably many boxes and that you can only fit countably many intersection point in each box, then you're done. :)

Maggie Miller - 5 years, 1 month ago

Ah, thanks. You can only get two intersection points in a sufficiently small box.

Mark C - 5 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$