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[5] viXra:2412.0128 [pdf] replaced on 2024-12-26 18:46:52
Authors: Joseph Musonda
Comments: 5 Pages.
Trisecting an arbitrary angle using a straightedge and compass only has been one of the oldest mathematical geometric problem tracing back to Euclidian times. This problem was never solved until 1837 when it was proven impossible by French Mathematician Pierre Wantzel. As stated by Pierre Laurent Wantzel (1837), the solution of the angle trisection problem corresponds to an implicit solution of the cubic equation x cubed minus 3x minus 1 equals 0, which is algebraically irreducible, and so is the geometric solution of the angle trisection problem. This method explained here can trisect any acute arbitrary angle. Only a compass and straightedge is used. The formal proof is later given after a practical illustration. For practical sake and to prove the possibility of trisecting an arbitrary angle, the author used the most common angle of 60 degrees that mathematicians uses to explain the proof for impossibility. The author believes that this proof will act as a basis for further research in geometry in future.Keywords: trisecting, arbitrary angle, geometry, straightedge and compas, implicit solution
Category: Geometry
[4] viXra:2412.0122 [pdf] replaced on 2025-11-11 01:21:47
Authors: Urs Frauenfelder, Joa Weber
Comments: 79 Pages. 7 Figures.
In this article we consider operators of the form ∂sξ + A(s)ξ where s lies in an interval [−T , T ] and s → A(s) is continuous. Without boundary conditions these operators are not Fredholm. However, using interpolation theory one can define suitable boundary conditions for these operators so that they become Fredholm. We show that in this case the Fredholm index is given by the spectral flow of the operator path A.
Category: Geometry
[3] viXra:2412.0118 [pdf] submitted on 2024-12-19 09:44:56
Authors: Hans Hermann Otto
Comments: 6 Pages.
The rainbow angle of about 42° is comparable with the golden mean based angle between edge and base of the Great Pyramid. It allows bringing together different areas of knowledge in an amusing way using simple geometry besides laws of optics. The mathematical exercise may encourage students to understand spectacles of nature in a simple and didactical manner.
Category: Geometry
[2] viXra:2412.0053 [pdf] submitted on 2024-12-09 21:22:33
Authors: Tai-Choon Yoon
Comments: 15 Pages.
Cycloid, semi-cycloid, elliptic cycloid, and elliptic semi-cycloid are all types of trochoids, and parts of roulette. They all refer to curves that trace their paths as they roll along a straight line, a circular orbit, or an elliptical orbit. Unlike cycloids, semi-cycloids are the curves traced by a point on a bicycle wheel as it rolls around the bicycle axle. Elliptic cycloids and elliptic semi-cycloids are the curves traced by ellipses as they roll along a straight line, but ellipses do not roll as smoothly as circles on a straight line. I also investigated the curves traced by circles and ellipses as they roll along circular or elliptical paths.
Category: Geometry
[1] viXra:2412.0015 [pdf] submitted on 2024-12-05 17:35:05
Authors: George William Tokarsky
Comments: 56 Pages.
We find a finite neighbourhood of the star flare (10,20).
Category: Geometry
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