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:
- https://www.youtube.com/user/njwildberger
- http://web.maths.unsw.edu.au/~norman/
- http://www.wildegg.com
- http://www.cut-the-knot.org/pythagoras/RationalTrig/CutTheKnot.shtml
- http://farside.ph.utexas.edu/euclid.html
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()
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.
Now, doing it again with random triangles (to show it works)
And again... this time with the liberty of using top-bottom hyperbolas
Bernoulli lemniscate, rotated around an axis in 3d (all rational points)
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
And again in antiview from http://antiprism.com
Descartes Kissing Circles.. but using hyperbolas instead of circles. ("Green Circles")
Note they are still tangent in some interesting places.
Hyperbolic sheet, rational points
Various Pythagorean Triple patterns
Sphere, Torus, rational points
Warped torus, created by slightly altering the rational parameterization
#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
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.