Comment by fwlr
21 hours ago
(Also aphantasic)
I think about the axis on which the fold is performed. Because of “square” and “diagonal”, for the two corners to touch, the fold axis must lie on both of the “other two corners”. A reflection about an axis does not change points on the axis, both points are unchanged, so their distance is also unchanged.
Yes that's correct. How do you answer this one: a hollow cube lying with one face on the horizontal like a die on a table has a hollow pyramid inside with the same base horizontal on the bottom like a usual pyramid. Inside the pyramid is as large a sphere as will fit. A horizontal ray is cast from outside of the cube through the entire structure, passing through each surface it would hit. The location of the ray is not specified, only that it is horizontal.
1) State the order of the surfaces it hits/passes through if you choose to cast it at a point where it passes through the maximum number of surfaces in the structure.
2) Is it possible to choose a different point where it would still pass through the cube but would pass through a different number of surfaces? (such as by missing the pyramid or sphere or both). State the distinct counts of how many surfaces it could pass through depending on where you cast it horizontally, like how many different counts exist. (Being tangent does not count as passing through a surface.)
I would answer this question by just picturing it and moving the ray around in my mind. How do you do it?
This is a bit more stream of thought, hopefully that is helpful.
The pyramid can’t take up all the space inside the cube, only another cube could do that. So there must be some points that are “inside cube, outside pyramid” and other points “inside cube, inside pyramid”. The pyramid is given as wholly inside the cube so there are no “outside cube, inside pyramid” points. Same for sphere inside the pyramid. So it’s
I struggle parsing the ray description, “horizontal” must meaningfully constrain the ray somehow (otherwise it wouldn’t have been mentioned), but I don’t know what that changes. Then I think about facts I know about rays - they can be uniquely specified with two points, or one point and one direction - oh, and we are given the direction is always “horizontal”, so we only need one point. The ray goes on forever so every “in” is “enter+exit”, so add 2 for every “in”. So it’s
1) Maximum, that’s 6, from the last option, so “cube, pyramid, sphere…”, have to exit the innermost first, so “…sphere, pyramid, cube”. I don’t know where this ray is positioned, “halfway” might be a good guess, or maybe pyramid is half of square and circle is half of pyramid so “a quarter way up”.
2) Distinct, that’s `0 2 4 6`. `0` doesn’t count because we want “still passes through the cube”, so 3 counts exist. Thankful that “tangents don’t count” so I don’t have to think about them, because nothing immediately comes to mind on how to handle them.
yes this is correct. Thanks for writing that out, it's really interesting to read how you solved this without picturing it.
It's a bit surprising to me that it was unclear what I meant by horizontal/level, I thought this word is well-known and easy to understand. Here's what I mean by horizontal: For me, it means that with respect to the ground, the ray isn't rising or falling as it travels, i.e. it's level (like a plane flying in level flight with respect to the ground, not rising or falling). This constraint doesn't really matter I guess, except that I said that the cube is also level on the ground on one of its surfaces, so since you have to cast the ray in a horizontal/level way, you can't cast the ray at an angle through one of the corners so it goes through the opposite corner of the cube, since - if you think of the ray like a plane - you can only move the angle of the ray side to side not up and down, it stays in level flight = horizontal, so you can't cast it through both opposite corners if the cube is lying flat on the horizontal on one of its surfaces. As I just wrote this paragraph, I pictured what my description of casting through both corners the would look like and why it's not level with the ground anymore. I guess it's harder to follow my words without picturing it.
You mentioned it's not clear how being horizontal constrains the ray, based on what I wrote and what I meant about it being level, is it now clearer how this constrains some of the rays that can be cast? (Like how you can't cast a ray through the opposite ends of the cube now)? Or is that hard for you to follow? For me I think maybe my words could be written in a bit confusing way but I think someone can understand them if they can picture what I referred to. Without that mental picture it could be difficult though.
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