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piliko

Piliko is a mish mash collection of highly experimental Geometry codes, mostly inspired by Rational Geometry. Most of the code tries to avoid irrational and transcendental numbers and functions as the basis of calculations.

#Current status

The code is bits and pieces, like a workshop or notebook, of various things. This code is 'alpha' level. It is highly experimental. It may or may not work at any time.

The main 'piliko' python code contains experimental implementations of Rational Geometry functions like spread, red, blue and green quadrance, and associated stuff like circumcenters of a triangle, antisymmetric polynomial, etc. The whole type system has not been thought out very carefully and many functions are partially or wholly unimplemented. Tests have not been created.

There are also a lot of random tidbits not related to the main piliko package, for example, some experiments:

  • Make rational number types that allow 0 as a denominator, in the Go language
  • Generate Sphere, Torus, tessellations of disks, Bernoulli Leminscate, etc, using only rational points, and also using Blue,Red,Green quadrance as short-hand notation for generation formulas.
  • Draw patterns of pythagorean triples
  • Find line-line intersection using the Wedge function of Geometric Algebra
  • Find conic equation of a spline, given 3 points, using Rob Johnsons "Conic Splines" paper of 1991, from Apple and http://sympy.org , a symbolic python mathematics package
  • Draw Ford Circles and Descarte's Kissing Circles, using Circles, then also Hyperbolas (red circles/green circles)
  • Show non-obvious facts about using floating-point IEEE numbers to model geometry, like the fact that the length of an object varies depending on its position in space, or that sometimes x+1.0 != x+1
  • Generate 3d OFF files of Ellipson shape (from Wildberger's Divine Proportions)
  • Geodesic sphere based on Icosahedron, using unusual algorithm

#Disclaimer

Rational Trigonometry was discovered and developed by Dr. Norman J Wildberger. His concepts and terminology are used here without permission and the use of these ideas/terms doesn't imply his endorsement nor his affiliation. All apologies if there are mistakes which are entirely this authors and not his. Please see these sites for more information:

The author of this code is not an expert.

#Piliko Examples

First, get python up and running, using any of the hundreds of tutorials from a web search. Then do this

from piliko import *

p1,p2,p3 = point(0,0),point(3,0),p3 = point(0,4)
print quadrance(p1, p2)
L1 = line( p1, p2 )
L2 = line( p1, p3 )
s = spread( L1, L2 )
print s

t = triangle(point(0,0),point(4,3),point(2,5))
oc,cc,nc = orthocenter(t),circumcenter(t),ninepointcenter(t)
print oc,cc,nc
print collinear( oc, cc, nc )

p1,p2,p3,p4 = point(0,0),point(3,0),point(2,0),point(6,0)
print is_harmonic_range( p1, p2, p3, p4 )

p=point(3,4)
bq,rq,gq=blue_quadrance(p),red_quadrance(p),green_quadrance(p)
print p,' ',bq,rq,gq,' ',sqr(bq),sqr(rq),sqr(gq)

To run files under example folder:

    export PYTHONPATH=. # linux only, not sure how to do on Windows
    python examples/example02.py

When the add-on 'matplotlib' package is installed on your system, you can do some basic plotting of pictures. For example:

t = triangle(point(0,0),point(4,3),point(2,5))
oc,cc,nc = orthocenter(t),circumcenter(t),ninepointcenter(t)
circ = circle( oc, blueq( oc, point(0,0 ) )
plot_circle( circ )
plot_points( oc,cc,nc )
plot_triangle( t )
plotshow()

Pictures

Reproducing a graph from Wildberger's Chromogeometry paper, 2008

This is program generated, using python functions to generate circle centers, hyperbolas, etc, using antisymmetric polynomials & other ideas from Wildberger's videos, papers, etc.

chromogeometry

Now, doing it again with random triangles (to show it works)

chromogeometry 2 chromogeometry 3

And again... this time with the liberty of using top-bottom hyperbolas

chromogeometry 4

Bernoulli lemniscate, rotated around an axis in 3d (all rational points) rational bernoulli 3d rotation

Ellipson, a 3d shape from Wildberger's Divine Geometry, created with the help of "Convex Hull" from http://antiprism.com and Meshlab from http://meshlab.sourceforge.net and the Renderer set to the "glass" Shader ellipson

And again in antiview from http://antiprism.com ellipson again

Descartes Kissing Circles.. but using hyperbolas instead of circles. ("Green Circles") Note they are still tangent in some interesting places. hyperbola version of Ford circles

Hyperbolic sheet, rational points another hyperbollic sheet

Various Pythagorean Triple patterns pythagorean triples pattern pythagorean triples pattern 2 pythagorean triples pattern 3 pythagorean triples pattern 4

Sphere, Torus, rational points rational points on a sphere torus with rational points

Warped torus, created by slightly altering the rational parameterization different shape of toroid, rational points

#Differences with real-number geometry

Geometry under rational numbers is different than 'ordinary' geometry using the real numbers. Here are some interesting and fun comparisons.

Real                               Rational
----                               --------

The real number line is            The Rational line is fundamental
fundamental and used               and irrationals are avoided.
in calculations

Points can have any                Points can only have Rational
coordinates                        coordinates

Distances are fundamental          Quadratic Forms (Quadrance)
sqrt(x^2+y^2)                      are fundamental. x^2+y^2.
                                   Distance is derived from Quadrance by 
                                   finding a square root, which is not
                                   always possible if it's not a perfect square

The point 1,1 is on                There is no circle centered at
a circle with center 0,0           0,0 containing the point 1,1 with a rational
                                   radius

Any line with points inside        Some lines with points inside
the circle intersects the          the circle won't intersect it
circle                             at rational points in x,y coordinates

Infinite series approximations     Finite bit-width numbers are 
are used in calculations.          considered the base of calculations

e + pi is considered as            e and pi are both infinite series
a number (of unknown nature)       so an attempt is made to avoid them

Angles are used to measure         Spread is used (sin squared)
line separation

Angles can be simply added.        Spreaded lines are combined using
                                   spread polynomials.

The basic right triangles are      Basic right triangles are the Pythagorean
45-45-90, 30-60-90, 60-60-60       triple triangles, like 3,4,5  5,12,13, etc. 
(angle based) with irrational      Many angle-based triangles have irrational 
lengths.                           points.

Wedge between vectors can be       Wedge between vectors is focused on
expressed with 'sin'               the rational calculation, x1*y2-x2*y1

Regular polygons are specified     Many regular polygons have irrational
by angles added.                   points and so can only be approximated

Hyperbolic, Elliptical,            Hyperbolic, Elliptic, and Spherical
Spherical are special cases        geometry share many of the same formulas
with very different formulas       and ideas.

Circles are described with         Circles are described using
sine and cosine approximations     rational paramterizations

Infinite series and sets are       Infinite series and sets are avoided
commonplace

Infinity is something we cant      Infinity can be dealt with using
deal with concretely               projective geometry (points-at-infinity map
                                   to finite points in 'ordinary' space, perhaps
                                   of a lower dimension)

Tangents can have any slope        Tangents have rational slope
                                   
Tangents exist at every point      Tangents exist at rational points


Copyright License

All computer code here is as of writing by Don Bright, github.com/donbright 2013-2016. Many formulas have been taken from other places, with attirbution in the code.

This code is free for use under a basic BSD-style Open Source Software license as described in the LICENSE file. You can freely copy and use it per the terms in that LICENSE file.

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Experimental geometry code, based on rational numbers

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