three dimensional angles

Higher-dimensional geometry (previously "Polyshapes").

three dimensional angles

Postby elpenmaster » Wed Mar 03, 2004 5:27 am

where two lines intersect in a polygon is an angle, so what about where two or more faces meet in a polyhedron at a line, or where some polyhedron meet in a polychoron? what are these called and how are they measured (in degrees?)?
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Postby Aale de Winkel » Wed Mar 03, 2004 6:27 am

At the points of intersection every object locally have tangential planes these planes have angles between them as defined in regular vector algebra.
cos(α) = V[sub]1[/sub] . V[sub]2[/sub] / (|V[sub]1[/sub]| |V[sub]2[/sub]|)
sin(α) = V[sub]1[/sub] * V[sub]2[/sub] / (|V[sub]1[/sub]| |V[sub]2[/sub]|)
α is the angle between the vectors, '.' is here the vectors inner product, '*' the vectors outer product, '|V|' the length of the vector.
The vectors here are penpendicular to the tangential planes.

The above is the regular vector algebra method, but defines only angle on a specific point, which can be different from point to point. So you'll get in general different values on every intersection point, it might be quite interesting to find correlations between those values given certain object.

When two n-spheres intersect the angles will be the same on every intersection point (merely the vectors point differently)
When two cones intersect, it might depend on wether the cone point is inside the other or not, and the relative angle between the cone.

I haven't the faintest of wether this is ever investigated, good luck if you try.
.
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spherical angles

Postby elpenmaster » Fri Mar 05, 2004 6:28 am

the measure of an angle is directly related to how much of a circle, that it is the vertex of, is inside of the angle. in a sphere, planes (or curved surfaces) extending from the center of the sphere would enclose a certain amount of the sphere. for example, a vertex of a tetrahedron has a measure in the percent of a sphere that the inside of the 3-d angle takes up. is there any formula for calculating this, the measure of a 3-d angle froma vertex?
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Postby Aale de Winkel » Fri Mar 05, 2004 10:49 am

no doubt about it, but I'm not someone who believes in coughing up formulae, try to find one yourself:

Take two circles with radi&iuml; R[sub]1[/sub] and R[sub]2[/sub] some distance d apart, of course d < R[sub]1[/sub] + R[sub]2[/sub] to have intersections.
Calculate the intersection points of the two circles (one would be enough).
Then calculate the two vectors from the circle centers to that intersection point, the angle between those two vectors gives the desired angle or is the desired angle complement (180-angle(?)) depending one what your definition is.

I do think this would give you an expression with the distance d and the two radi&iuml; as parameters which should give you 180[sup]o[/sup] when d = R[sub]1[/sub] + R[sub]2[/sub], other obvious values might also be easily given.

Do try this yourself first, math is fun when you get the hang of it, I'll try this procedure myself this weekend, but as said if you can do it yourself you can calculate the result you want for other objects.

happy formulae hunting :lol:
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Postby elpenmaster » Tue Mar 09, 2004 4:37 am

maybe the measure of the 3-d angle divided by 360 would = area of intercepted surface of sphere divided by four pi r squared
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Postby chitspa » Wed Mar 17, 2004 8:48 pm

I think that the easiest way to veiw an angle in 4-D would be to imagine a cone. Then to give the cone a 3-D coordinate system. The angles of the cone can be measured in the x and y directions. ie. the measurement of an angle in 4-D can be expressed as 30x,45y. Although this isn't a perfect cone since it has an oval base it is still an angle in 4-D.
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Postby pat » Wed Mar 17, 2004 10:02 pm

chitspa wrote:I think that the easiest way to veiw an angle in 4-D would be to imagine a cone. Then to give the cone a 3-D coordinate system. The angles of the cone can be measured in the x and y directions. ie. the measurement of an angle in 4-D can be expressed as 30x,45y. Although this isn't a perfect cone since it has an oval base it is still an angle in 4-D.


I think that approach has some merits, but it causes problems if the base of the cone is other than elliptical. The more usual approach (mathematically) is to draw a small sphere centered at the vertex of the angle. Calculate the proportion of the surface of that sphere which is contained in the angle. Multiply that proportion by the surface area of a unit sphere in that number of dimensions and you have the angle in radians (or steradians or hyperradians or...).
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Postby elpenmaster » Sat Mar 20, 2004 5:08 am

so a plane is basically the 3-d equivalent of a straight angle
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Postby pat » Sun Mar 21, 2004 12:11 am

elpenmaster wrote:so a plane is basically the 3-d equivalent of a straight angle


Indeed. But, it would not necessarily be the only solid angle with that measure.
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Postby elpenmaster » Sun Mar 21, 2004 5:35 am

if you took a regular tetrahedron and put the vertex in the center of it, the angle of one of the sides would be 90 degrees, for a cube it woud be 60 degrees. in a regular polytope, the center angle to one of the lines, or sides, or cells, would be 360 divided by number of sides.
right?
:D
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Postby pat » Sun Mar 21, 2004 7:07 am

elpenmaster wrote:if you took a regular tetrahedron and put the vertex in the center of it, the angle of one of the sides would be 90 degrees, for a cube it woud be 60 degrees. in a regular polytope, the center angle to one of the lines, or sides, or cells, would be 360 divided by number of sides.
right?
:D


Sure, except that solid angles aren't usually measured in degrees... but if they were... and if all of space were 360 of them, then sure.
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Postby elpenmaster » Sun Mar 21, 2004 7:15 am

what are solid angles measured in? how many of them are the in a sphere?
:?
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Postby pat » Sun Mar 21, 2004 5:16 pm

elpenmaster wrote:what are solid angles measured in? how many of them are the in a sphere?


If you look back at my March 17th post, you'll see that I mentioned that n-dimensional angles are measured by the amount of surface area they would enclose on a unit sphere centered at the vertex.

In 2-d, these are radians. And there are 2π radians in a full circle (S<sup>1</sup>). Thus, an angle that contains 3/4-ths of the unit circle centered at its vertex has a measure of 3π/2 radians.

In 3-d, these are steradians. And there are 4π steradians in a full unit sphere (S<sup>2</sup>). Thus, an angle that contains 3/4-ths of the unit sphere centered at its vertex has a measure of 3π steradians.

In n-d, these units don't have a name that I know of so I'll say n-radians. Since the n-dimensional sphere (S<sup>(n-1)</sup>) has surface area: f(n) = <sup>2<sup>(n+1)/2</sup>π<sup>(n-1)/2</sup></sup>/<sub>(n-2)!!</sub> for odd n and f(n) = <sup>2π<sup>n/2</sup></sup>/<sub>(n/2 - 1)!</sub> for even n (where k!! = k * (k-2) * (k-4) * (k-6) ... 5 * 3 * 1), then there are f(n) n-radians in a full unit hypersphere (S<sup>(n-1)</sup>). Thus, an angle that contains 3/4-ths of the unit hypersphere centered at its vertex has a measure of 3f(n)/4 n-radians.

So back a few posts then, an n-dimensional hyperplane would be the equivalent of an n-dimensional flat angle. It would have measure f(n)/2 n-radians. For n > 2, there would be lots of angles not made by a single hyperplane which would also measure f(n)/2 n-radians. If I have time later, I'll try to work out one in 3-d. But, I think if you start with a pyramid with a rectangular base and always keep the internal angle between two opposite planes a fixed (non-unit) proportion of that of the internal angle between the other two opposite planes, you will be able to make a solid angle of 2π steradians that isn't flat.

Note: I editted this post and it replaced all of my π's with question marks. I think I fixed them all, but if there are stray question marks still, know that I meant π.
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Postby elpenmaster » Thu Apr 01, 2004 1:00 am

why is it that in 2-d, the circumference of a circle divided by the diameter equals a very long number (3.14159265358979323846264338327950288419716939927510. . .)
but in 3-d space, the surface area of a sphere divided by the area of a great circle in it is a very simple "4"?
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Postby pat » Thu Apr 01, 2004 5:30 am

elpenmaster wrote:why is it that in 2-d, the circumference of a circle divided by the diameter equals a very long number (3.14159265358979323846264338327950288419716939927510. . .)
but in 3-d space, the surface area of a sphere divided by the area of a great circle in it is a very simple "4"?


It's somewhat funky. The circle in 2-d is called S<sup>1</sup>. The ratio of S<sup>2k+2</sup> to S<sup>2k+1</sup> is always a rational number. The ratio of S<sup>2k+1</sup> to S<sup>2k</sup> is always a rational mutliple of pi. *shrug*
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Postby Aale de Winkel » Fri Apr 02, 2004 10:22 am

This funky business might on a somewhat phylosophical level have to do with the fact that we have a linear number system (ie based on the line). This makes the ratio of the circles circumference a multiple of an irrational number π (in fact 2πR).
Since all hyper-spheres have the same factor in there properties, those cancel out upon division into a rational number.

I gather this makes the circle stand out from the line the way it thus, question now rises given a line with length R, the circle relating to this line has length 2πR. When I take some line with length (2π)[sup]2[/sup]R let ends meat, does this object relate in the same way to the circle as the circle relates to the line, especially when we had a circular based number system?

Note, I haven't the faintest clue how a non-linear number system looks like, Sci-fi freaks (like me) might remember the metamorph on moon-base alpha, she had an elliptical based number system and calculated things faster then the moonbase computer :lol:
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Postby pat » Fri Apr 02, 2004 5:39 pm

Aale de Winkel wrote:I gather this makes the circle stand out from the line the way it thus, question now rises given a line with length R, the circle relating to this line has length 2?R. When I take some line with length (2?)[sup]2[/sup]R let ends meat, does this object relate in the same way to the circle as the circle relates to the line, especially when we had a circular based number system?


I have no idea what you mean here, but I think it has merit! :D

Geometrically, we're assuming that space is flat. Its curvature is zero. If space had positive curvature, then the circumference of a circle would be less than 2πR. If space had negative curvature, then the circumference of a circle would be more than 2πR.

For that matter, we're also assuming the Euclidean metric. If we use the L<sub>1</sub> norm (also known as the Taxi-Cab Geometry), then the circumference of a circle is 4R. The surface area of the sphere is 4R<sup>2</sup>. (Assuming, I did my calculations correctly.)
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Postby Polyhedron Dude » Sat Apr 17, 2004 7:48 am

elpenmaster wrote:why is it that in 2-d, the circumference of a circle divided by the diameter equals a very long number (3.14159265358979323846264338327950288419716939927510. . .)
but in 3-d space, the surface area of a sphere divided by the area of a great circle in it is a very simple "4"?


The surface content of the n-sphere is n x pi^(n/2) x r^(n-1) / (n/2)! - ! denotes factorial.

The content of the internal n-1-sphere is pi^((n-1)/2) x r^(n-1) / ((n-1)/2)!

The ratio of the first to the second will be n x sqrt(pi) x ((n-1)/2)! / (n/2)!.

According to the gamma function, (1/2)! = sqrt(pi/2) and this is a factor for other half factorials such as (3/2, 5/2, etc) (3/2! = 3/2 times 1/2!). When n is odd the 1/2! will be in the denominator and will cancel with the sqrt(pi) factor in the numerator and therefore lead to a rational number. When n is even the 1/2! will be in the numerator where it combines with the sqrt(pi) factor turning it into a rational multiple of pi. Thats why that happens.

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Postby elpenmaster » Sun Apr 18, 2004 5:36 am

does that mean that the surface of a glome divided by a great sphere in the glome will be an irrational number?
What is this number?
:? :P
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Postby pat » Sun Apr 18, 2004 5:15 pm

elpenmaster wrote:does that mean that the surface of a glome divided by a great sphere in the glome will be an irrational number?
What is this number?
:? :P


Using the formulae from mathworld, I come up with (3/4)?.

By my understanding, the surface content S<sub>n</sub> of the unit n-sphere (where this is the sphere in n-dimensions) is <sup>2π<sup>(n/2)</sup></sup>/<sub>(n/2)!</sub>. The volume of the unit (n-1)-sphere is <sup>S<sub>n-1</sub></sup>/<sub>(n-1)</sub>.

This leaves the ratio as <sup>(n-1) √π ((n-1)/2)!</sup> / <sub>(n/2)!</sub>. Note: this differs slightly from Polyhedron Dude's formula above. Additionally, according to mathworld, (3/2)! = (√π)/2.
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Postby Aale de Winkel » Fri May 14, 2004 8:56 am

With respect to the Γ function you guys are a bit confusing:
Γ(n) = (n-1)! so Γ(3/2) = (1/2)! = ( √π) / 2
as Γ(1/2) = √π

see: http://mathworld.wolfram.com/GammaFunction.html
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Postby Rybo » Sun May 16, 2004 4:15 am

elpenmaster wrote:why is it that in 2-d, the circumference of a circle divided by the diameter equals a very long number (3.14159265358979323846264338327950288419716939927510. . .)


Pi is not a long number. A long number is finite. Pi is not a finite number Pi is numerical-like reference to infinity, beginning at the number three an rising from there. I know this is being picky but Im lousy at math and geometry while at same time being interested in what mysterys of Universe can be unfolded using geometry.
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Postby Keiji » Sun May 16, 2004 7:44 am

No, pi is not infiity, it is an irrational number. Meaning that it has infinite decimal places. Say, I wonder how calculators calculate the value of pi to any number of decimal places...
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Postby Rybo » Mon May 17, 2004 12:39 am

Aale de Winkel wrote:Note, I haven't the faintest clue how a non-linear number system looks like, Sci-fi freaks (like me) might remember the metamorph on moon-base alpha, she had an elliptical based number system and calculated things faster then the moonbase computer :lol:


Aale, I too looked into this with my geometric discoveries of the hexagonal pattern of prime numbers ergo 2-D exploration but have always gone blank with 3-D numerical study of prime patterns specifically using the four planes of the cubo-octahedron.
Anyway I wanted to share with you and others this link to some geometric studies using 5-fold geometries to define a point-like space.
http://www.codefun.com/Geometry_quantum.htm

It is called Rafki.
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Postby wendy » Wed Jan 19, 2005 5:51 am

The units i use for the angles are.

For all dimensions, the totality of space, measured as 1, and written in base 120.

For 2D, also, the degree, there being 360 in a circle

For 3D, also, the degree excess, there being 720 to all-space.

For measures in radians, the prismatic and tegmatic radians are used, the ratio of these are 1 prismatic radian = n! tegmatic radian.

The solid angle of the simplex lies between 1 and sqrt(n/e) tegmatic radians, e=2.71828182846&c.

The solid fraction of 4-space, occupied by the six platonic polychora are

(all space = 120s, 1s = 120 f, f are divided into 120 parts)

x3o3o3o 5-choron 1s 20.9f
x3o3o4o 16-choron 5s
o3o3o4x 8-choron 7s 60f aka 'tesseract'
x3o4o3o 24-choron 15s 0f
o3x4x3o 48-choron 30s 50f aka 'octagonny'
x3o3o5o 600-choron 33s 15.5f aka 'fifhundchoron'
x5o3o3o 120-choron 38s 24f aka 'twelftychoron'

It should be noted that the solid angle at the vertex of a twelftychoron is very nearly 1/pi. pi = dec 3.14159265359 = 3:16E8 E3 1/pi=0:3823.80

The last three have a solid angle at the vertex greater than that of any duoprism.
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