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If you inscribe one triangle on a spherical system, you inevitably describe four triangles. There is a concave small triangle and a concave big triangle, as viewed from inside, and a convex small triangle and a convex big triangle, as viewed from outside. Concave and convex are not the same, so at minimum there always are inherently four triangles.
Background Nothingness: One spherical triangle ABC drawn on the Earth's surface inadvertently produces four triangles as the corners of the surface triangle are inherently related to the center of the Earth D, and their lines of interrelatedness together with the three edge lines of the surface triangle describe a tetrahedron. (See Fig. 812.03.) Drawing a triangle on the surface of the Earth (as described at Sec. 810) also divides the surface of the Earth into two areas__one large, one small__both of which are bound by a closed line with three edges and three angles. The large triangle and the small triangle have both concave and convex aspects__ergo, four triangles in all. Euler did not recognize the background nothingness of the outside triangles. (See Sec. 505.81.)
Under the most primitive pre-time-size conditions the surface of a sphere may be exactly subdivided into the four spherical triangles of the spherical tetrahedron, each of whose surface corners are 120-degree angles, and whose "edges" have central angles of 109 28'. The area of a surface of a sphere is also exactly equal to the area of four great circles of the sphere. Ergo, the area of a sphere's great circle equals the area of a spherical triangle of that sphere's spherical tetrahedron: wherefore we have a circular area exactly equaling a triangular area, and we have avoided use of pi .
Originally posted by reficul
ok, so what about phi - the golden ratio? was davinci stoned too?
poussin? the architects of gothic cathedrals?!
perhaps the builders of the ancient pyramids didn't recognize pi the way we do today but the pyramids are right in front of us to proove the theory!
Using infinity as a constant when obtaining finite measurements is just fubared.
Further, having an imaginary number at the core of your discipline is no more valid than having an imaginary deity at the core of your theology.
Originally posted by Eidolon23
And that right there is the problem: perfect circles do not exist, except in the human mind.
How many triangles do you think you need to not use pi and get that accuracy?
The most perfect manmade spheres constructed to date are the fused solid quartz gyroscopic rotors built for NASA's Gravity Probe B spacecraft. There are four spheres onboard, each measuring 3.81 cm (1.5 in) across. Their average departure from mathematically perfect sphericity is 1.8 X10-7 of the diameter. This means that, if scaled up to the size of the Earth, the maximum height/depth of topographic features would be 1.5 m (4.92 ft).
Oh it's dense alright, in this meaning of the word:
Originally posted by Eidolon23
There are many mathematicians who are advocating the replacement of pi with tau (another irrational number, but one that allows one to work more easily with real-world geometries). Here's an excellent, if dense article:www.math.utah.edu...
a : slow to understand : stupid, thickheaded