Naked Science Forum
General Science => General Science => Topic started by: f.point on 27/05/2015 11:04:09
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Solution trisecting (n-sections) and n-proper construction of the polygon using divider and straightedge.
The ball shall be awarded in two equal parts, obtained half contains two areas, the circle (represents a planar geometry) and a semi-sphere (represents spherical geometry), the circle is the boundary between the circuit and semi-sphere
View photo (below)
CIRCLE
within a given arbitrary angle BAC,straightedge (straightedge is flexible, it can draw on a sphere)
straight line the BA to extend the circle to give the point D
SPHERE
straightedge connect points B and D, you get the curve BD
straightedge and divider - a procedure divisions curve into two equal parts is the same as the process of division shall exceed the level in two equal parts, we get the point E
straightedge - connect points C and E and get the curve CE
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Proportion longer exists in plane geometry, I found that the process can be applied to the sphere
choose point G
divider EC, from point G we get a point Hdivider EC, from point H we get a point I
divider EC, from point E twe get a point Jdivider EC, from point J get the point K
divider EC, from point K get the point L
straightedge point L and point I and connect, we get curve LI
EG divider, from the point L we get a point P
EG divider, from the point P and get the point Ostraightedge join the dots E and P and proceed to the circle, we get the point Q
straightedge merge points E and O and proceed to the circle, we get the point R
CIRCLE
straightedge connect point A and point Q, we get straight line AQ
straightedge connect point A and point R, we get straight line AR
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these have carried out the production of a given trisection angle, is obtained from the rest of the (n-section, a regular n-polygon) ...
Now proclaim everywhere that I decided 2- millennium math problems
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I had a quick skim, but it's a long time since I did any geometry...
straightedge is flexible, it can draw on a sphere
Traditionalists will have a problem with this step; Euclid's style of geometry was originally done in a "Euclidean" space (ie not curved).
However, there is a modern equivalent to the straight line in a non-Euclidean geometry: The geodesic (http://en.wikipedia.org/wiki/Geodesic). On a sphere, all geodesics are segments of great circles.
straightedge point L and point I and connect, we get curve LI
I have a slight problem with this step. Curve LI is like lines of latitude, and is not a geodesic, so it can't be considered a "Straight line in a curved geometry", or drawn with a straight edge (even a bendy one).
I guess you could draw LI easily enough with a compass...
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Traditionalists will have a problem with this step; Euclid's style of geometry was originally done in a "Euclidean" space (ie not curved).
in the rules says that you should use an unmarked straightedge and dividers, I respect the rules, no one next to me did not find that the proportion of arc can be solved in sphere,
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Is trisection of a triangle considered ethical? Triangles have rights too you know! :-)
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No!
1. Since EG is arbitrary, you could select EG such that it is not a rational fraction of the great circle. Then the entire construction is invalidated.
2. A chord on a great circle subtends a smaller angle than the same chord on a small circle.
Sorry, mate, but the proof that an angle cannot be trisected by Euclidean construction, remains supreme.