Python, Ipython and Jupyter

Sage is based on the Python language, which is very popular (web programming, graphical interaces, scripts, ...) and easy to learn.

As of version 8.3, Sage uses Python 2 and is in a phase transition towards Python 3.

The way you will mostly interact with Sage is through IPython which is an enhanced Python interpreter. Sage and IPython can be used in two modes: in a console or through Jupyter. This document is an example of a Jupyter worksheet.

sage: S = {17*n for n in range(10) if n%2 == 1}
sage: S

sage: 124 in S

sage: sum(S)

sage: {3*i for i in S}

sage: 8324074213.factor()

sage: m = matrix(ZZ, 3, 3, [0,3,-2,1,4,3,0,0,1])

sage: m.eigenvalues()

sage: m.inverse()

sage: R.<x> = PolynomialRing(ZZ, 'x')  # ZZ = ring of integers

sage: R

sage: P = 6*x^4 + 6*x^3 - 6*x^2 - 12*x - 12
sage: P.factor()

sage: P2 = P.change_ring(QQ)     # QQ = field of rational numbers

sage: P2.factor()

sage: P3 = P.change_ring(AA)     # AA = field of real algebraic numbers

sage: P3.factor()

sage: P4 = P.change_ring(QQbar)  # QQbar = field of complex algebraic numbers

sage: P4.factor()

sage: in

sage: inc

sage: inf

sage: gamma?

sage: f(x) = sin(x)^2 -sin(x)
sage: f

sage: f.in

sage: G = grap

sage: # edit here

sage: G.vertex

sage: # edit here

Calculator

Integration (symbolic):

sage: integral(e^(-x^2), x, -Infinity, Infinity)

sage: integral(1/sqrt(1+x^3), x, 0, 1)

Integration (floating point numeric)

sage: numerical_integral(1/sqrt(1+x^3), 0, 1)

Integration (certified numeric with arbitrary precision). This example would only work if you have a Sage version >= 8.2

sage: R = ComplexBallField(128)
sage: R.integral(lambda x,_: 1/(1+x^3).sqrt(), 0, 1)

sage: R = ComplexBallField(1024)
sage: R.integral(lambda x,_: 1/(1+x^3).sqrt(), 0, 1)

Computing roots

sage: f(x) = x^5 - 1/3*x^2 - 7*sin(2*x) + 1

sage: plot(f, xmin=-2, xmax=2)

sage: r1 = find_root(f,-2,-1)
sage: r1

sage: r2 = find_root(f,0,1)
sage: r2

sage: r3 = find_root(f,1,2)
sage: r3

sage: plot(f, xmin=-2, xmax=2) + point2d([(r1,0),(r2,0),(r3,0)], pointsize=50, color='red')

Latex:

sage: M = Matrix(QQ, [[1,2,3],[4,5,6],[7,8,9]]); M

sage: latex(M)

sage: M.parent()

sage: latex(M.parent())

Some 3d Graphics:

sage: x, y = SR.var('x,y')
sage: plot3d(sin(x-y)*y*cos(x), (x,-3,3), (y,-3,3))

Interaction:

sage: var('x')
sage: @interact
sage: def g(f=sin(x)-cos(x)^2, c=0.0, n=(1..30),
....:         xinterval=range_slider(-10, 10, 1, default=(-8,8), label="x-interval"),
....:         yinterval=range_slider(-50, 50, 1, default=(-3,3), label="y-interval")):
....:     x0 = c
....:     degree = n
....:     xmin,xmax = xinterval
....:     ymin,ymax = yinterval
....:     p   = plot(f, xmin, xmax, thickness=4)
....:     dot = point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))
....:     ft = f.taylor(x,x0,degree)
....:     pt = plot(ft, xmin, xmax, color='red', thickness=2, fill=f)
....:     show(dot + p + pt, ymin=ymin, ymax=ymax, xmin=xmin, xmax=xmax)
....:     pretty_print(html('$f(x)\;=\;%s$'%latex(f)))
....:     pretty_print(html('$P_{%s}(x)\;=\;%s+R_{%s}(x)$'%(degree,latex(ft),degree)))

Licence and development

SageMath is distributed under a GPL licence which means that you can freely download the software, have access to its source code and you can redistribute it in any form you like as long as you use a GPL-compatible licence.

The source code access can be done with two question marks ?? directly in a cell

sage: gamma??

The Sage project began in 2005 under the inpetus of William Stein and is now being developed by hundreds of developers around the world. Most of the development is happening on the trac server and the sage-devel mailing list.

Extra packages for geometry and dynamics

There are several packages built on top of Sage dedicated to geometry and dynamics. We will study them in more depth during this Sage Days 96. Let us mention

• flipper: mapping classes (homeomorphisms

  of surfaces)

• snappy: 3-d hyperbolic geometry
• surface_dynamics: interval exchange transformations, origamis and more
• flatsurf: translation surfaces (affine transformation, linear flow, etc)

These packages are not installed by default in Sage. The instructions to install them are available on the wiki https://wiki.sagemath.org/days96

Let us use flipper and check the braid relation on a surface of genus 2 with 1 puncture. The surface S_2, 1 (that is builtin in flipper) is depicted on the picture below

Here is how to play with the Dehn-twist around the curves a and b

sage: import flipper

sage: S = flipper.load('S_2_1')
sage: a = S.mapping_class('a')
sage: b = S.mapping_class('b')
sage: print(a*b*a == b*a*b)   # braid relation
sage: print(a*b == b*a)       # these do not commute

With snappy installed you can investigate 3-dimensional hyperbolic manifolds. It comes with an extensive database of them. Here we compute some invariantes of the manifold "m015" from the database.

sage: import snappy

sage: M = snappy.Manifold("m015")
sage: M

sage: M.cusp_info()

sage: M.alexander_polynomial()

sage: M.volume()

sage: M.complex_volume()

Flipper can be used to construct mapping tori of pseudo-Anosov homeomorphism and send them to snappy for further analysis

sage: import flipper
sage: import snappy

sage: S = flipper.load('S_2_1')
sage: a = S.mapping_class('a')
sage: b = S.mapping_class('b')
sage: C = S.mapping_class('C')
sage: d = S.mapping_class('d')
sage: f = a * b * C * d
sage: f.nielsen_thurston_type()

sage: M = snappy.Manifold(f.bundle())
sage: M.volume()

With surface_dynamics installed you can play with origamis (an origami is a finite cover of a square torus ramified at most over the origin)

sage: import surface_dynamics

sage: o = surface_dynamics.Origami('(1,2)', '(1,3)')

sage: o.stratum()

sage: o.plot()

sage: V = o.veech_group()
sage: print V
sage: print V.nu2(), V.nu3(), V.ncusps()

sage: V.farey_symbol().fundamental_domain()

Finally, flatsurf allows you to construct translation surface from polygons and play with translation flow. Below we construct saddle connections on the double pentagon

sage: import flatsurf

sage: S = flatsurf.translation_surfaces.veech_double_n_gon(5)
sage: S.plot()

sage: sc = S.saddle_connections(20)
sage: S.plot() + sum(s.plot(color='red') for s in sc)

Flipper pseudo-Anosov can also be sent to flatsurf as follows

sage: import flipper
sage: import flatsurf

sage: S = flipper.load('S_2_1')
sage: a = S.mapping_class('a')
sage: b = S.mapping_class('b')
sage: C = S.mapping_class('C')
sage: d = S.mapping_class('d')
sage: f = a * b * C * d
sage: S = flatsurf.translation_surfaces.from_flipper(f)

The essentials

• Use Tab completion to browse and access documentation with ?
• The main website: http://www.sagemath.org/ (including some HTML documentation)
• A forum to ask your questions about Sage: http://ask.sagemath.org
• A book "Calcul mathématique avec Sage"/"Computational Mathematics with SageMath"/"Rechnen mit Sage", a book about Sage (in french, english and german): http://sagebook.gforge.inria.fr/

What's next?

What's next?
----------

Go to the wiki https://wiki.sagemath.org/days96 and choose a worksheet. If you just start with Sage or does not know much about Python programming, it would be a good idea to work on the 6 Programming worksheets ("First steps with Sage", "Learn about for loops", etc).

You can also have a look at the Sage documentation. It can be accessed from a Jupyter notebook by clicking on "Help". Or from the main website http://www.sagemath.org/.