PyGeo

a Dynamic Geometry Laboratory

In mathematics, as in any scientific research, we find two tendencies. On the one hand, the tendency toward *abstraction* seeks to crystalize the *logical* relations inherent in the maze of material that is being studied, and to correlate the material in a systematic and orderly manner. On the other hand, the tendency toward *intuitive understanding* fosters a more immediate grasp of the objects one studies, a live *rapport* with them, so to speak, which stresses the concrete meaning of their relations.

As to geometry, in particular, the abstract tendency has here led to the magnificant systematic theories of Alegraic Geometry, of Riemannian Geometry, and of Topology; these theories make extensive use of abstractreasoning and symbolic calculations in the sense of algebra. Notwithstanding this, it is still as true today as it ever wasthat *intuitive* understanding is of great value in geometry. And such concrete intuition is of great value not only forthe research worker, but also for anyone who wishes to study and appreciate the results of research in geometry…..

For it is true generally speaking, that mathematics is not a popular subject , even though its importance may be generally conceded. The reason for this is to be found in the common superstition that mathematics is but a continuation, a further development, of the fine art of arithmetic, of juggling with numbers. Our book aims to combat that superstition by offering instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader can construct. This book was written to bring about a greater enjoyment of mathematics, by making it easier for the reader to penetrate to the essence of mathematics without having to weight himself down under a laborious course of studies.

David HilbertPreface

Geometry and the Imagination (programming and the imagination?)June, 1932

Geometry and the imagination

Hello Point

>>> from pygeo import *

Visual-2003-07-09

>>> window=display(axis=True)

>>> Point()

Point[0 0 0]

Hello Line

>>> from pygeo import *

Visual-2003-07-0

>>> window=display(axis=True)

>>> p1=Point()

>>> p2=FreePoint(1,-7,4)

>>> Line(p1,p2)

LineFromPoints(Point[0 0 0],FreePoint[1 -7 4])

>>> window.pickloop()

VPython (visual)• The rendering engine for PyGeo

• http://www.vpython.org

The Visual library is Copyright (c) 2000 by David Scherer.

David Scherer wrote VPython while a sophomore at CMU

• Bruce Sherwood, physics professor now at NC State, was successful in achieving NSF funding for the continuing development of VPython. Jonathan Brandenmayer, a NC State Comp Sci student just released the Boost version of VPython in beta.

• New version will link to Numarray, rather than Numeric, new and supported GTK widget for OpenGL.

• cylinder; arrow; cone; sphere; ring; box; curve; convex; label

• VPython is cool!

.. I advocate here, as always in such general lectures, a tendency which I like best

to designate by the phrase "fusion of arithmetic and geometry"—meaning by arithmetic,

as is usual in the schools, the fields which includes not merely the theory of integers,

but also the whole of algebra and analysis.

In fact it has long been the custom in the schools as well as the university,

first to study the geometry of the plane and then, entirely separated from

it, the geometry of space. On this account, space geometry is unfortunately

often slighted, and the noble faculty of space perception, which we posses orignally,

is stunted.

Felix Klein

Introduction

Elementary Mathematics From an

Advanced Standpoint

Volume II - Geometry

1908

3d is better than 2d

Readibility counts SUI vs GUI

• Cognitive involvement and learning. “Hard fun!” … Alan Kay• Designing a forgiving script user interface for non-programmers. • Factory function implementation.• The help - a resource of resources.

Code as Application

Geometry without algebra is dumb! – Algebra without geometry is blind!

David Hestenes David Hestenes Clifford Algebra to Geometric CalculusClifford Algebra to Geometric Calculus19841984.

The intention is:

The code of PyGeo is as much the application as the graphics. And therefore….. READIBILITY COUNTS

from pygeo import *

v=display(scale=55,axis=0,panel=1,observe_on=1)

p1 = FreePoint(10.,-7,3,label='p1',color=BLUE,fontsize=10)p2 = FreePoint(-3,9,4,label="p2",color=WHITE)p3 = FreePoint(7,-14,3,label="p3",color=WHITE)Triangle(p1,p2,p3,style=FILL)

v.pickloop()

Dynamic is better than static

Twenty years ago the study of projective geometry would have seemed somewhat like the study of Latin: good for the brain and a means to keep in contact with the ancients. Today, however, projective geometry is a popular subject with mathematicians and computer scientists. This is due to the new applications to computer vision and to the fact that geometry and geometric thinking are again on the upswing. Moreover, computer science has returned the favor by providing powerful tools for the visualization and diffusion of mathematics. Java applets, animated GIF images, and linked HTML pages allow us to present mathematics and, most particularly, geometry in a way that was unthinkable twenty years ago.

Introduction to Projective Geometry Inaugural Lecture

Juan Carlos Álverez Paiva