How common is this property in geometry? I know that fractals like the Koch Snowflake also have infinite perimeter over finite area, but I don't know what else does.
Doesnt seem that uncommon.
IgorPartola is perfectly right to mention geometric series, you can easily use a geometric progression to construct a shape with infinite perimeter and finite area, e.g. by gluing together rectangles with height one and width decreasing in geometric progression. With a bit more thought you can also construct a smooth shape having this property.
The geometric series sums to 2 - your glued together rectangles will have perimeter 2*(1+2) and area 2*1.
No. You should think through that perimeter calculation one more time, preferably while drawing a picture.
Here's a hint: the perimeter of a rectangle is no less than its height; you can glue so that the perimeter of each rectangle contributes at least 1 to the perimeter of the union.
First, a rectangle of height 1 and width 1/2. The perimeter is 1 * 2 + 1/2 * 2, two sides of height 1 and two sides of width 1/2.
You "glue" the second rectangle. As one may understand this, you glue them by putting them one beside the other standing up, i.e. you glue them along one of the heights. Sorry for the crude ascii art:
---- -- ------
| | || | |
| | || | |
| | + || -> | |
| | || | |
| | || | |
---- -- ------
Go on doing that and for a rectangle of width 1/n, you only add 2 * 1/n to the perimeter. In the end you get a single rectangle with height 1 and width 2. The perimeter is 2 * 1 + 2 * 2.
---
Now, maybe, you may want to specify that you glue the rectangles along their widths, not their heights.
That way, the resulting shape when you add the second rectangle is not a rectangle but an irregular shape with 6 sides. Sorry for the crude ascii art again:
1
----------
| |n/2
| | 1
n| ----------
| |n/2
| |
--------------------
2
But you didn't specify this option over the other one. And, honestly, if we talk about putting rectangles in a sequence, I think it's more common to think of the rectangles as standing up side by side with their heights together as in the first option. For the second option I would describe the rectangles as having a fixed width of 1 and decreasing heights.
(It's Friday night people it's a joke and I have no idea what the article is talking about just looked at the picture)
To all the flat earthers out there, this property can be used to find out earth is not flat, just by drawing a giant triangle on the surface, without leaving the earth. Historically, to prove the earth is round, people have relied on the sun shining directly overhead on wells in different cities. But this approach proves it without the need to refer the sun.
That wasn't to prove the Earth is round (and it doesn't prove it). Eratosthenes assumed two things when he performed his experiment: 1) the Earth is round, and 2) the Sun is an infinite distance away. By just this experiment he would have been unable to distinguish between this situation and the Earth being flat while the Sun being only a finite distance overhead (and in fact a fair bit closer than it actually is). Eratosthenes and his contemporaries were already convinced of the roundness of the planet, and he simply wanted to measure it.
>But this approach proves it without the need to refer the sun.
A flat-earther would just tell you that you're not able to maintain a straight path over such long distances without relying on external guides that would definitely put you on curved paths. If the Earth is flat and you stand at 0 N 0 E, how do you move in a straight line East of there? I.e. continuously moving towards the South because the polar coordinates curve towards your left as you progress.
This is something that was more or less solved a long time ago with surveying instruments. You don't have to move in a straight line, you build triangles out of sight lines.
EDIT: Also, that's to measure distance, not direction.
https://www.noaa.gov/digital-collections/collections/4774/it...
https://amerisurv.com/2012/10/27/the-longest-line/
And measurements of, say, very precise equilateral triangles, necessarily imply certain interior angles, which you can compare to the actual angles they make. For instance, on a flat plane, you can fit six equilateral triangles sharing one point to make a hexagon. On a sphere if you make them big enough you'll find that they don't quite fit.
Like I said before, you can't do that over water. It constrains the shape of the triangle you can measure.
Is the idea that land is spherical while the oceans are flat? Do this survey on each continent, including Antarctica, again just constructing big triangles and measuring the deviation from flatness. Eventually your model of the flat earth looks like a bunch of pieces of an eggshell separated by flat water that just happens to assemble into an egg- or rather an ellipsoid- if rearranged, up to and including the expected flattening at the poles.
With sufficient elbow grease you could extend a survey from the far north of the Americas down to the south, measuring the curvature as you went, eventually finding that you must be almost upside-down compared to where you'd started.
Do flat-earther reject the existence of LASER, too?
Only if you're happy "proving" your argument to an audience that never had any doubts. You can't use this argument to prove the earth is not flat over the objections of your audience because you can never convincingly show that any given line is straight.
To expand on that, it’s about community and finding people who share your interests. The movie Behind The Curve explores this idea and it’s quite revealing.
It's implausible, yet that's what stimulates the tribal feelings among the believers.
Makes me think that mr trump switched from being democrat to republican and pushed for magaesque folks who often love him to the death due to very similar principle - just spit out some populist crap that stirs core emotions - the worse the better, make them feel victim, find easy target to blame which can't defend themselves well (immigrants), add some conspiracy (of which he is actually part of as wall street billionaire).
Extreme left wouldn't swallow easily that ridiculous mix from nepotic billionaire who managed to bankrupt casinos and avoided military duty (on top of some proper hebephilia with his close friend mr E and who knows what else).
But what do I know, just an outside observer, but nobody around the world has umbrella thick enough that this crap doesn't eventually fall on them too.
I've heard that even Hitler was like this: that he didn't start out hating Jews, but repeatedly reacted to the fact that he got louder cheers whenever he blamed things on Jews. But I don't know how to verify if this is true.
https://www.thehistoryreader.com/historical-figures/hitlers-...
Unfortunately, the climate change deniers in all their forms have made it much further by having support in politics and having a real impact on people's lives. In contrast to flat earthers.
Just the mere fact that my post here could be interpreted as political (which it really isn't) is evidence of this.
Jeran from Behind The Curve was one of the ones to flip, and since then, he's been making videos on how the earth is actually round.
He has a lot of thoughts on what it actually takes to convince other flat-earthers. I found it somewhat interesting: https://www.youtube.com/watch?v=1grMf17PeEk
It was just one city actually. The critical piece is that the city's northern latitude was nearly identical to the Earth's angle of axial tilt. Which also means that this shadow phenomenon only occurs during the Summer Solstice.
https://www.khanacademy.org/science/shs-physical-science/x04...
Rough estimate - with an excellent 0.5" angular resolution and 35km triangle this could work.
Either they are overlapping which violates the definition of a triangle, or they don't and the parallel lines always maintain the same distance X to each other and consequently maintain distance X at infinity (let's say X=1, bc you can just scale it).
dkdcio•2mo ago
I don’t follow, how/why?
dadoum•2mo ago
ironSkillet•2mo ago
roywiggins•2mo ago
That edge is basically an artifact of the model, you can equally model the hyperbolic plane space as a disk and then the boundary is a circle, or on an actual hyperboloid in 3D and it extends out forever.
itishappy•2mo ago
jerf•2mo ago
If that sounds like so much technobabble, that's because this article assumed what I think is a very specific level of knowledge about hyperbolic space, as it doesn't explain what it is, yet this is one of the very first things you'll ever learn about it. So it has a rather small target audience of people who know what hyperbolic space is but didn't know that fact about triangles. If you'd like to catch up with what hyperbolic space is, YouTube has a lot of good videos about it: https://www.youtube.com/results?search_query=hyperbolic+spac... And as is often the case with geometry, videos can be a legitimate benefit that is well taken advantage of and not just a "my attention span has been destroyed by TikTok" accomodation.
Including CodeParade's explanations, which are notable in that he made a video game (Hyperbolica) in which you can even walk around in it if you want, with an option for doing it in VR (though that is perhaps the weirdest VR experience I had... I didn't get motion sick per se, but my brain still objected in a very unique manner and I couldn't do it for very long). It's been out and on Steam for a while now, so you can run through the series where he is talking about the game he is in the process of creating at the time and go straight to trying it out, if you want.
volemo•2mo ago
Accidentally, I’m in that small set: I have a hand-wavy understanding of hyperbolic spaces (the high school I went to was named after Lobachevsky!), but I haven’t studied the geometry and didn’t know the formulae for area.
gus_massa•2mo ago
You can use a map that is inside a circle with r=1. The objects get deformed, but points have a 1 to 1 correspondence. Lines that pass though 0 look straight, but other lines are curved.
Measuring a distance is hard, you have to use some weird rules.
If you draw a segment of length 0.001 segment in the circular map, it has almost the same length in the real infinite map.
If you draw a segment of length 0.001 segment near the border of the circular map, it's a huge thing in the infinite map.
Moreover, a line that pass thorough 0 has apparent length 2 in the map, but represent an infinite length in the plane
Note that the border of the circle is outside the plane.
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The reverse happen if you have a map of the Earth. You can draw on the map with a pencil a long segment near the pole, but it represents a small curved segment in the Earth.
---
Back to your question ,,,
It's on the hyperbolic plane, not in the usual euclidean plane. So the map is only the top half, and the horizontal line = axis x is outside, it's the border.
Length are weird, and a 0.001 segment draw with a pencil on the map far away from the x axis is small in the actual hyperbolic plane, but a 0.001 segment draw with a pencil on the map near the x axis is very long in the actual hyperbolic plane.
The circles "touch" the x axis. In spite they look short when you draw them with a pencil, they part that is close to the x axis has a huge length in the hyperbolic plane.
kazinator•2mo ago
We can make an analogy to cartography: you can't trust areas and distances on distorted projections like Mercator.
Look, even the angles don't look to be zero in that diagram. We have to imagine that we zoom in on an infinitesimal zone around each corner to see the almost zero angle; i.e. the circle tangent lines actually go almost parallel. So to speak.
Thus the angles are locally correct, since they are measurable on arbitrarily small scales and can easily be imagined to be even when glancing at the entire figure. But distances between the points aren't localizable; they have to follow a measure which somehow correctly spans the abstract hyperbolic space that they represent.
How about this (almost certainly incorrect) imagining: pretend that the real line shown, on which the three points lie, is actually a horizon line, which lies in a vast distance (out at infinity). Just like the horizon when you do drawings with two-point perspective. Imagine the three points are vanishing points on the horizon. Vanishing points are not actually points; they just directions into infinity.
if, in a two-point perspective, you draw a curve whose endpoints are tangent to two vanishing point traces, that curve is infinitely long.
For instance if you draw an intersection between two infinite roads, where the curb has a round corner, you will get some kind of smiley curve joining two vanishing points. That curve is understood to be infinitely long.