Divine proportions: Rational trigonometry to universal geometry

2006 ◽  
Vol 28 (3) ◽  
pp. 73-74
Author(s):  
James Franklin
Keyword(s):  
Rational Trigonometry ◽  
Universal Geometry

scholarly journals The Feuerbach Theorem and Cyclography in Universal Geometry

KoG ◽  
2020 ◽  
pp. 47-58
Author(s):  
William Beare ◽  
Norman Wildberger
Keyword(s):  
Finite Field ◽  
Unit Circle ◽  
Finite Fields ◽  
Algebraic Approach ◽  
Rational Points ◽  
Field Case ◽  
Rational Trigonometry ◽  
Universal Geometry ◽  
Simplified Form

We have another look at the Feuerbach theorem with a view to extending it in an oriented way to finite fields using the purely algebraic approach of rational trigonometry and universal geometry. Our approach starts with the tangent lines to three rational points on the unit circle, and all subsequent formulas involve the three parameters that define them. Tangency of incircles is treated in the oriented setting via a simplified form of cyclography. Some interesting features of the finite field case are discussed.

scholarly journals Rational Trigonometry in Higher Dimensions and a Diagonal Rule for $2$-planes in Four-dimensional Space

KoG ◽  
2017 ◽  
pp. 47-54
Author(s):  
Norman Wildberger
Keyword(s):  
Dimensional Space ◽  
Higher Dimensions ◽  
Natural Generalization ◽  
Principal Angles ◽  
Rational Trigonometry ◽  
Pythagoras Theorem ◽  
The Cross

We extend rational trigonometry to higher dimensions by introducing rational invariants between $k$-subspaces of $n$-dimensional space to give an alternative to the canonical or principal angles studied by Jordan and many others, and their angular variants. We study in particular the cross, spread and det-cross of $2$-subspaces of four-dimensional space, and show that Pythagoras theorem, or the Diagonal Rule, has a natural generalization forsuch $2$-subspaces.

scholarly journals THE ASYMPTOTIC STRUCTURE OF SPACE-TIME

2003 ◽  
Vol 12 (09) ◽  
pp. 1743-1750 ◽  
Author(s):  
FRED C. ADAMS ◽  
MICHAEL T. BUSHA ◽  
AUGUST E. EVRARD ◽  
RISA H. WECHSLER
Keyword(s):  
Dark Matter ◽  
Vacuum Energy ◽  
Space Time ◽  
Dark Matter Halo ◽  
Constant Density ◽  
Dark Matter Halos ◽  
Asymptotic Structure ◽  
Density Profiles ◽  
Near Future ◽  
Universal Geometry

Astronomical observations strongly suggest that our universe is now accelerating and contains a substantial admixture of dark vacuum energy. Using numerical simulations to study this newly consolidated cosmological model (with a constant density of dark energy), we show that astronomical structures freeze out in the near future and that the density profiles of dark matter halos approach the same general form. Every dark matter halo grows asymptotically isolated and thereby becomes the center of its own island universe. Each of these isolated regions of space-time approaches a universal geometry and we calculate the corresponding form of the space-time metric.

scholarly journals The Universal Geometry of heterotic vacua

2019 ◽  
Vol 2019 (2) ◽  
Author(s):  
Philip Candelas ◽  
Xenia de la Ossa ◽  
Jock McOrist ◽  
Roberto Sisca
Keyword(s):  
Universal Geometry

scholarly journals GARIS DI LAPANGAN HIMPUNAN BILANGAN BULAT MODULO 17

2013 ◽  
Vol 5 (1) ◽  
pp. 1
Author(s):  
Denni Hariati Sinaga ◽  
Idha Sihwaningrum ◽  
Ari Wardayani
Keyword(s):  
Unique Property ◽  
Rational Trigonometry

n this paper we discuss rational trigonometry in the field F17 ,in particular point, lines and their properties. A unique property in this field is given by the null lines.

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