3 ms·

I understand the optical fiber version, but the fractal version went flying right over my head. Can anyone give a simpler explanation?

valarauca1 12y ago
Think of the mask like windows in a wall. As the sun moves (or as the earth spins) the light from each window will change its angle. So if we add another wall with more windows, we can filter our filtered light. Then you'd say, "when the light shines in window 4 it's say 6pm". Now add a board person with a degree in mathematics, and a grid of windows. And you get single "pixels/windows" that show light based on the minute and hour, and BOOM a digital clock.
wetmore 12y ago
tl;dr If, for each angle of the sun, you want some object to have a particular shadow, that object exists. The theorem says something like this: imagine you have a 2D plane, with the usual x and y axes. For any θ between 0 and pi, there is a unique line passing through the origin. That's L_θ [1]. Now for each line, say we pick out some subset of the line, called G_θ [2]. In the picture I've made G_θ thicker for emphasis, but if you think of the whole line as a bunch of points, then G_θ is just some of those points. So for each θ, we have a line, and a corresponding subset of the line. If we add all the lines together we get the entire 2D plane, and if we add all the G_θ's together we get some set of points in that plane. The important condition in the theorem that needs to hold for the results to be true is that if we add all the G_θ's together, we get a set which we can say has some area. Now proj_θ F is sort of like the "shadow" of F on L_θ, for some set F. See this picture [3]. The perpendicular projection takes a 2D set and projects it onto a 1D set (the line). Analagously, if the sun was in the sky above your head and you were standing on a 2D plane, then your shadow would be the perpendicular projection of a 3D set (you) onto a 2D set (the ground). Anyway, if we add all the G_θ's together and get some set which is suitably "nice", then there is some other 2D set F such that if we project F onto any* line L_θ, the "shadow" on L_θ covers G_θ completely, and the part of the shadow that isn't covering G_θ is negligibly small [4]. So applied to the sundial, this means there exists some shape such that its shadow at some time of day will be that time. * Technically, it's "almost any", which means, informally, for all but a negligible number of lines. The stuff about measure, measurable and almost all is all from measure theory [5]. I can explain some of the measure theory concepts if you'd like. [1] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#0 http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#0 [2] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#1 http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#1 [3] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#2 http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#2 [4] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#3 http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#3 [5] http://en.wikipedia.org/wiki/Measure_(mathematics) http://en.wikipedia.org/wiki/Measure_(mathematics)
- drob 12y ago
  Very nice! That's a thoroughly earned upvote. Measure theory is super cool.
  - wetmore 12y ago
    Thanks, I really appreciate the compliment. And yeah, it's great.

valarauca1 12y ago

Think of the mask like windows in a wall. As the sun moves (or as the earth spins) the light from each window will change its angle. So if we add another wall with more windows, we can filter our filtered light. Then you'd say, "when the light shines in window 4 it's say 6pm". Now add a board person with a degree in mathematics, and a grid of windows. And you get single "pixels/windows" that show light based on the minute and hour, and BOOM a digital clock.

wetmore 12y ago

tl;dr If, for each angle of the sun, you want some object to have a particular shadow, that object exists. The theorem says something like this: imagine you have a 2D plane, with the usual x and y axes. For any θ between 0 and pi, there is a unique line passing through the origin. That's L_θ [1]. Now for each line, say we pick out some subset of the line, called G_θ [2]. In the picture I've made G_θ thicker for emphasis, but if you think of the whole line as a bunch of points, then G_θ is just some of those points. So for each θ, we have a line, and a corresponding subset of the line. If we add all the lines together we get the entire 2D plane, and if we add all the G_θ's together we get some set of points in that plane. The important condition in the theorem that needs to hold for the results to be true is that if we add all the G_θ's together, we get a set which we can say has some area. Now proj_θ F is sort of like the "shadow" of F on L_θ, for some set F. See this picture [3]. The perpendicular projection takes a 2D set and projects it onto a 1D set (the line). Analagously, if the sun was in the sky above your head and you were standing on a 2D plane, then your shadow would be the perpendicular projection of a 3D set (you) onto a 2D set (the ground). Anyway, if we add all the G_θ's together and get some set which is suitably "nice", then there is some other 2D set F such that if we project F onto any* line L_θ, the "shadow" on L_θ covers G_θ completely, and the part of the shadow that isn't covering G_θ is negligibly small [4]. So applied to the sundial, this means there exists some shape such that its shadow at some time of day will be that time. * Technically, it's "almost any", which means, informally, for all but a negligible number of lines. The stuff about measure, measurable and almost all is all from measure theory [5]. I can explain some of the measure theory concepts if you'd like. [1] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#0 http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#0 [2] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#1 http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#1 [3] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#2 http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#2 [4] http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#3 http://imgur.com/dx15rLx,7xB9tpv,a3evQpR,35j9cwu#3 [5] http://en.wikipedia.org/wiki/Measure_(mathematics) http://en.wikipedia.org/wiki/Measure_(mathematics)

drob 12y ago
Very nice! That's a thoroughly earned upvote. Measure theory is super cool.
- wetmore 12y ago
  Thanks, I really appreciate the compliment. And yeah, it's great.