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# Beautiful, intriguing, elegant ideas Topic Page: 1 2

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I am curious, is the pythagorean theorem or some version of it applicable to spherical trig (triangles on the surfaces of spheres and other shapes)?

For a right angle triangle composed of geodetics on a sphere with radius R, the Pythagorean theorem takes a very elegant form:
cos(c/R) = cos(a/R).cos(b/R).

Johannes--

Few astronomical observations have really changed our perspective on earth's place in the universe. Those that did were made around 1610 with a 20x power, 0.3° field of view telescope, sporting a singlet objective stopped down to 15 mm.

I mostly use small, fast, and (ultra)portable glass: Bushnell Legend 8x42, Pentax PCF WP II 20x60, MiniBorg 60ED, ST-80, EON-80, ST-120, 6" StarBlast.

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If a MÃ¶bius strip is going to be included, shouldn't a Klein bottle be too?

'Topology' is about mapping one shape onto another(and hopefully back), same applies to the triangle onto sphere, any shape onto another, there's more than one way around the barn though I think the branches (algebraic or point set) are converging (if not considered already so). Relatively simple questions like this can be astonishingly difficult to prove - as was the recently solved Poincare Conjecture. It's important to consider exactly which space(flat, curved, other) you are mapping to and from and backwards upside down. It's in many ways important to consider if asking for all potential sizes or small from one to another. Small triangle onto large Klein bottle, Moebius stip, sphere... or large onto small... helps if creating a Klein bottle 'opener'

Clear skies - Jim Crazy PNW imager !

A-P Mach1 on 12" concrete pier - ROR structure.

Tak FSQ85-EDX, Tak FS-128. Orion 8" f/3.9 w/MoonLite motorized - MPCC MkIII CCD:

SBIG 8300M/FW8, Astrodon 36mm LRGB, 5nm Ha, 3nm SII, OIII - Canon EOS 6D unmod SSI3, SSAG,

AA Flat-Man

Skytools3pro, MaxImDLpro, PSCS5, PSPpro, TheSkyX, TheSky6, BYE, StarTools

Orion XX14g -for visual- diags, ep's, accy tubes, Binocs .

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If they ever launch a warp-drive spaceship, I hope they break a Klein bottle across the bow.

"Scientists aren't perfect, just peer reviewed."
"Eye of Sauron Observatory", featuring "Sauron's Other Eye", 16" dob, conical Royce mirror.

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The drive will be powered by a 4D hamster running on a Mobius treadmill. (Dilithium crystals - ha!)

- Rick

Cactus Patch Observatory / 14" LX200

"The four points of the compass be logic, knowledge, wisdom, and the unknown. Some do bow in that final direction. Others advance upon it. To bow before the one is to lose sight of the three."

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