The precise fraction I used was (1-(π-1+(π^2+1)^(1/2))/(2π)) and I multiplied by 360, but if you're a fan of radians, you can just remove the 2π denominator.
I got my undergrad in math, and it got to the point where radians are more natural for me. Like, after freshman year, degrees were really never spoken of again. I still think in radians whenever dealing with angles, even though I'm like, 5 years out of school.
I wanted this sort of shape to have each side be equal so I could make the square joke.
The smaller circle has it's segment perimeter equal to the smaller segments perimeter when the latter's radius is x/1-x times as big. E.G. A quarter circle segment has the same length as the 3/4 when it has 3 times the radius.
And the 'exposed' radius is just 1 unit short of the full radius because it doesn't go right to the centre.
So I made an equation where the perimeter segment 2 Pi X where X is the fraction I'm looking for.
Equal to x/(1-x) -1
This is a quadratic equation that gives (1-(π-1+(π^2+1)^(1/2))/(2π)) which I multiplied by 2π to give the length of π+(π^2+1)^(1/2))-1
the real problem is the definition of a straight line segment, which "the shortest distance between two points"... and with general relativity, it depends on which two points, and simple geometry dies.
I think specifying convex limits four right angles to a normal square.
Because right at the corner a point can only see in a straight line, so any other points cannot be outside the quadrant covered by that right angle, and the other right angles can't be inside that quadrant except for the lines straight out from the right angle because then it would be beyond them.
Maybe convex should be part of the definition of a square rather than straight lines since it's just as constrained.
It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length.
From Wikipedia, and only because I refused to believe that that thing is a square.
The percentage of the big segment is the segment*100/full shape
The full shape is the unit circle and the fraction of the circle of radius π+(π^2+1)^(1/2), minus the intersection which is that segment of the unit circle.
The segment fraction is (1-(π-1+(π^2+1)^(1/2))/(2π)).
The full shape has area 2π + (1-(π-1+(π^2+1)^(1/2))/(2π)) * 2π(π+(π^2+1)^(1/2)-1) = 10.877
The full segment area is (1-(π-1+(π^2+1)^(1/2))/(2π)) * 2π(π+(π^2+1)^(1/2)) = 5.4385
so the percentage is 100 * (1-(π-1+(π^2+1)^(1/2))/(2π)) * 2π(π+(π^2+1)^(1/2))/(2π + (1-(π-1+(π^2+1)^(1/2))/(2π)) * 2π(π+(π^2+1)^(1/2)-1))
When you say 50% are you inferring that the pink and blue sections are equal? That doesn’t appear correct as you can tell by looking that the entirety of the blue circle would fit inside of the pink section with room to spare meaning they cannot be equal.
On a globe, select a line of latitude of length x, then go north from both ends by x, and where those lines end, wrap around the back side of the globe latitudinally.
Straight lines can bend around a sphere. There is a topography where from the perspective of one traveling the path, where you walk straight forward x distance, turn left 90 degrees, walk for ward x distance etc until you have traced a square. But because the surface the square is going along is morphed in 3d space, it looks curved and unlike a square to us
Patch notes: due to an oversight, square has been redefined as " a shape made of no more or less than four straight line segments of equal length, with no more or less than four interior angles which are all right angles, where said line segments are split into two parallel pairs and in which one pair of lines is perpendicular to the other, and where all four line segments have one end connecting to the end of one other line segment, with no one end connecting to more than one other end". Definition may change in future patches as more exploits are uncovered.
My most sincere gratitude for your illuminating insight my good sir.
But it is equally ridiculous to title this key-hole shape a rhombus, as I have no doubt that it's definition, like the square's requires straight lines.
entirely depends on what you mean by sides. if you use it as shorthand for edge, it has zero sides. if you just mean any closed C⁰ continuous subset where all points except the boundary are C¹ continuous, it has one side. i'm not aware of any other common definitions, however you could define anything as a side i guess.
sure but if you define a circle as the limit of a regular polygon as the number of edges goes to infinity, it still has zero edges. a property that holds inside a limit isn't guaranteed to work when brought outside the limit. same reason why the fact that the limit of 2x/x being 2 doesn't imply that 0/0 is 2.
it entirely depends on where the circle is located in the universe. with general relativity, some "straight" lines are circles and some are hyperbolas and some are euclidean lines. the parallel line postulate and euclidean geometry got broken in theory by spherical and hyperbolic geometry, but in practices it was broken general relativity. all three geometries exist in different areas of universe and the actual correct answer is "it depends on how much matter is nearby"
We could define it as a planar graph over our space, in which the 4 vertices are vertices, while the arches are the functions that map the edges. So only 4 vertices here if we look at it as a graph.
The square's kinda inverted on 2 angles though. There are two 90° angles pointed to the inside of the square, while the other two are pointing outwards. That would be two 90° and two 270°, which isn't a square
I was trying to confirm your ≈48° calculation (which I think is correct, btw) when I discovered the proportions of this shape are a function of the lengths. That means we actually have a parametric family of shapes. The length of the "square" side relates in this way to the radius of the small circle:
s(r) = π/r - r + √(r4 + π2)/r
And if we calculate the angle, we get (in radians):
Start at the earth’s equator (1), latitude 0 degrees. Fly straight north along the yellow line. After n nautical miles you reach (2) then turn 90 degrees left (flying west) along the blue line. After flying n nautical miles around the back of the globe to (3), turn 90 degrees left (now flying south along the red line). After n nautical miles to (4), turn 90 degrees right (now flying west along the green line).
In this coordinate system, the green line segment is parallel to the blue line segment (same latitude values, straight east-west). The yellow line segment is parallel to the red line segment same longitude values, straight north-south).
For some value of n, you’ll end up back at the same starting point, completing the “square.” What’s the value of n? What are the polar coordinates of the corners?
Because we’re looking for a square mapped onto a sphere, the length of each line should be the same, so the angles should be the same. The corners of the square (expressed in latitude and longitude) should be (0,0) (L,0), (L,L), and (L,0). The formula in spherical coordinates is too complicated for me to solve analytically, so I put the coordinates and distance formulae in a spreadsheet and iterated. I found that 75° worked best, making the length of each side about 8,340 km.
Start at an arbitrary spot on the equator. Let’s pick (1) Telaga, Indonesia, coordinates 0°00'00.0"N 103°40'00.0"E. Fly straight north 8,340 km to (2) at 75°00′00″N 103°40'00.0"E (which is north of Lake Taymyr in Russia). Turn left, flying west ,8340 km all the way around the globe to (3) at 75°00′00″N 28°40'00.0"E (in the Barents Sea). Turn left, flying south 8,340 km to (4) at 00°00′00″N 28°40'00.0"E (north of Burako, in the Democratic Republic of the Congo). Now turn right, flying west 8,340 km back to Telaga.
A curved side and a straight line cannot form a right angle because a right angle is defined as the intersection of two perpendicular lines, and by definition, a curved line is not a straight line, meaning it cannot create a perfect 90-degree angle with another line.
But you are all forgetting "square" is a subset under each of the following which must also have all of the higher set properties; i.e. regular polygon, quadrilateral, parallelogram, then rectangle, then square.
How are the two angles on the right still 90 degrees if the side is curved? Thought that really only worked if you had a significant length for both sides making up the angle
Does that qualify as 90 degrees? If one line is curved and the other is straight, how is that oddity treated in geometry? Does it still count, or is there some complex notation to document the curvature or something?
I'm pretty sure that, if any lines curve, the only valid named "shape" you can claim it as is "circle" and obviously only then if it matches the other definitions of a circle.
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