Pythonic Math

Pythonic Mathematics Kirby Urner EuroPython 2005 Göteborg, Sweden

Principal Themes Math Through Programming Math Through Storytelling Beyond Flatland Curriculum as Network

Math Through Programming Common goal: Literacy Levels of fluency Reading (Recognition) Writing (Recall) Planning Designing Testing and Correcting (Debugging)

Math Through Storytelling © USPS

Curriculum as Network: Typical Urner-style graph of inter- connecting topics

OO, Functions, More OO We need a big picture of the OO design at first, to make sense of dot notation around core objects such as lists i.e. “these are types, and this is how we think of them, use them.” Use lots of analogies, metaphors. Write your own classes a little later, after you’ve defined and saved functions in modules. Get more specific about syntax at this point.

From a class for home schoolers, taught by me @ Free Geek in PDX See: http://www.4dsolutions.net/ocn/pygeom.html

OO & Data Structures Aristotle and the Kingdoms (family trees) The Tree (big in classical thinking) Sets (big in mid 20 th century “new math”) Data Structures: list, array, vector, matrix, table, n-tuple, record, dictionary, hash, stack, queue, network, tree, linked list, graph… Python: a good language for show and tell plus plays well with others

New Hybrid: CS + Mathematics Mathematics as extensible type system A focus on algorithms that use objects Mix traditional notation w/ OO’s dot notation Math Objects: polynomials, integers, integers modulo N, permutations, vectors, matrices, quaternions, polyhedra … Python’s operator overloading: a gateway to new levels of abstraction

Python plays well with others The prospect of getting to do computer graphics is a motivational incentive to tackle “the hard stuff.” The promise: we’ll take you somewhere fun.

Types of Graphics with sample apps & libraries From presentation given at OSCON in 2004. See: http://www.4dsolutions.net/oscon2004/ POV-Ray VPython PyOpenGL POV-Ray Tk wxWidgets PyGame PIL Spatial Flat Still Moving

Jython + POV-Ray + QuickHull3D

Python + PIL http://www.4dsolutions.net/ocn/fractals.html

Python + Tk Cellular Automata ala Wolfram generated using graphics.py by John Zelle

Functions and Figurate Numbers def tri (n): return n * (n + 1) // 2 >>> [tri(x) for x in range (1, 10)] [1, 3, 6, 10, 15, 21, 28, 36, 45] Front cover: The Book of Numbers by Conway & Guy

Sequence Generators >>> def tritet(): term = trinum = tetranum = 1 while True : yield (term, trinum, tetranum) term += 1 trinum += term tetranum += trinum >>> gen = tritet() >>> [gen.next() for i in range (6)] [(1, 1, 1), (2, 3, 4), (3, 6, 10), (4, 10, 20), (5, 15, 35), (6, 21, 56)] from Synergetics by RBF w/ EJA

Polyhedral Numbers Animation: growing cuboctahedron >>> gen = cubocta() >>> [gen.next() for i in range (6)] [(1, 1, 1), (2, 12, 13), (3, 42, 55), (4, 92, 147), (5, 162, 309), (6, 252, 561)]

Cubocta Shell = Icosa Shell def cubocta (): freq = shell = cubonum = 1 while True : yield (freq, shell, cubonum) shell = 10 * freq**2 + 2 cubonum += shell freq += 1 From Synergetics Java + POV-Ray

Pascal’s Triangle def pascal (): row = [1] while True : yield row this = [0] + row next = row + [0] row = [a+b for a,b in zip(this, next)] >>> gen = pascal() >>> gen.next() [1] >>> gen.next() [1, 1] >>> gen.next() [1, 2, 1]

Fermat’s Little Theorem def gcd (a,b): """Euclidean Algorithm""" while b: a, b = b, a % b return abs(a) if isprime(p) and gcd(b,p) == 1: try: assert pow(b, p - 1, p) == 1 except: raise \ Exception, 'Houston, we’ve got a problem.'

Euler’s Theorem def tots (n): return [i for i in range (1,n) if gcd(i, n)==1] def phi (n): return len(tots(n)) if gcd(b,n) == 1: try: assert pow(b, phi(n), n) == 1 except: raise \ Exception, 'Houston, we’ve got a problem.'

RSA def demo (): """ Abbreviated from more complete version at: http://www.4dsolutions.net/satacad/sa6299/rsa.py """ plaintext = "hello world" m = txt2n(plaintext) p,q = getf(20), getf(20) # two big primes N = p*q phiN = (p-1)*(q-1) e = 3 s,t,g = eea(e, phiN) # Extended Euclidean Algorithm d = s % phiN c = encrypt(m,N) # pow(m, e, N) w/ booster newm = decrypt(c,d,N) # pow(c, e*d, N) plaintext = n2txt(newm) return plaintext

Math Objects: grab from the library and/or build your own >>> mypoly = Poly([ (7,0), (2,1), (3,3), (-4,10) ] ) >>> mypoly (7) + (2*x) + (3*x**3) + (-4*x**10) >>> int1, int2 = M(3, 12), M(5, 12) # used in crypto >>> int1 * int2 # operator overloading 3 >>> int1 – int2 10 >>> tetra = rbf.Tetra() >>> bigtetra = tetra * 3 # volume increases 27-fold >>> bigtetra.render()

Example: Build A Rational Number Type From http://www.4dsolutions.net/ocn/python/mathteach.py def cf2 (terms): """ Recursive approach to continued fractions, returns Rat object >>> cf2([1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]) (4181/2584) >>> cf2([1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]) (665857/470832) >>> (665857./470832) 1.4142135623746899 """ if len(terms)==1: return terms.pop() else: return Rat(terms.pop(0),1) + Rat(1, cf2(terms))

Conclusions “ Programming to learn” is as valid an activity as “learning to program.” My proposal is about programming in Python in order to build a stronger understanding of mathematics. Mastering specialized “learning languages” or even “math languages” doesn’t offer the same payoff as mastering a full-featured generic computer language. This approach and/or curriculum is not for everyone. A wide variety of approaches exist even within the broad brush strokes vision sketched out here.

Thank You! And now… HyperToon demo + Q&A Presentations repository: http://www.4dsolutions.net/presentations/

Pythonic Math

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