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I've came up with the idea of 3d angles and other n-dimensional angles. It can be represented by a n-dimensional sphere's surface area sector where the surface area is divided into spherical squares (360 of them) into n degrees (squared degrees, cubed degrees, etc.). There is no necessity for direction like 2d angles because that is merely a result of the fact that it is limited to go into only two directions positive and negative (of course with different magnitudes). I've found use in these structures because they helped me prove the distance formula for n dimensions. I have looked online to see if anyone has invented these but it came up with no results. Has anyone came up with this because it seems pretty obvious? If not where can I publish my idea? Here is a very abstract and rough draft of why they are useful

N-dimensional angles are a slice of an n-dimensional sphere instead of a circle sector. It angles are represented by a dividing a sphere into 360 squares on the surface. This is analogous to the surface area of an n-dimensional sphere. It doesn't have to have a specific direction like regular angles do because the only reason they have them is because the can only go in two directions positive and negative by a certain amount (this is merely a property of there low dimension). These can be use in numerous proofs such as the extended distance formula. Because every distance has right angle when a line is drawn perpendicular to the plane below the other two have to add up to 90 degrees. This even works for higher dimension with these angles because they are just projections of 2d angles into 3d dimensions similar to how the volume of a height of 1 is equal to the area of the top of the same figure. The reason these are similar is because angles are relative to distance and thus all angles represent a unit angle on the same line so the area of two unit 2d angles are equivalent to a unit 3d angle. As a side note angles determine the area taken up by their dimensions because they are linear predictions of the future. That being said, since all angles predict the future of area and two angles must add up to the right angle then those areas must add up to the right angle area. These are also useful in calculations such as the current rotation of a plane at a given moment because you can use three 2d angles (pitch, yaw, and roll) and multiply then to encode three angles in one.

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    $\begingroup$ Have you looked at en.wikipedia.org/wiki/Steradian ? $\endgroup$ – Zach Stone Jun 13 '15 at 21:18
  • $\begingroup$ I haven't but it seems like those only apply to three dimensions. $\endgroup$ – Jackson H Jun 13 '15 at 21:20
  • $\begingroup$ True, but the idea works in more. See en.wikipedia.org/wiki/… $\endgroup$ – Zach Stone Jun 13 '15 at 21:22
  • $\begingroup$ Looks interesting (and complicated); my calculation are a little simpler. Do you think I could publish this or is it too similar to this? Mine are divided into squares instead of circle which allows for the covering of the whole circle and with a little spherical geometry can be easily manipulated to the desired calculation. Such as finding the area under an angle (integral of angle) or the area produced from it (in the case of the extended distance formula). $\endgroup$ – Jackson H Jun 13 '15 at 21:25
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    $\begingroup$ You can't cover a sphere with uniform little "squares". $\endgroup$ – David K Jun 13 '15 at 22:56

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