Tree Gaps and Orchard Problems - Numberphile

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Strong rec Graviton · 2020-08 · Mathematics Stack Exchange · on “When you can look through a forest?”

There's a highly-related Numberphile video called Tree Gaps and Orchard Problems [this video] . I highly recommend taking a look. You're definitely not the only one with such curiosities!

Recommends Larry B. · 2018-01 · Mathematics Stack Exchange · on “Prove that any point I pick on the plane will lie on a line defined by integer coordinates and $(0,0)$?”

Here's a proof, which lulu hinted at in the comments: Suppose I have the point $(\pi, \sqrt{2})$, and there were a line $y = mx $ that it lies on. That line would need slope $m = \frac{\sqrt{2}}{\pi}$. What lattice point can we choose with that kind of slope? We can't choose any, since, for all lattice points $(a,b)\in\mathbb{Z}^2$, the resulting slope is a rational number $\frac{b}{a}\in\mathbb{Q}$, and $m$ is irrational. You might have tried a simpler proof, where "the lattice points go on forever", by noticing that the line $y = mx$ maps the real line onto the plane. The number of lines can be enumerated; the space of lines has the same cardinality as $\mathbb{R}^1\times\mathbb{N}^2$, whi

Recommends badroit · 2020-06 · Mathematics Stack Exchange · on “Intuitive, possibly graphical explanation of why rationals have zero Lebesgue measure”

With respect to "with very little background in math", this video might help to achieve a related intuition: [this video] ?