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Using Wolfram Language in Jupyter: A free alternative to Mathematica

In this post I am going to describe how to add the Wolfram Language to the Jupyter notebook. This provides a free alternative to Mathematica with, pretty much, the same syntax. The use of the Wolfram Engine is free for non-production as described in their website:

The Free Wolfram Engine for Developers is available for non-production software development.

You can use this product to:

  • develop a product for yourself or your company

  • conduct personal projects at home, at school, at work

  • explore the Wolfram Language for future production projects


To install you should do the following steps:

  • Download Wolfram Engine.

  • Create a Wolfram account, if you don't have one.

  • Execute the installer.

  • Type the following in a terminal


and you should be asked for your email and password.

After that you should be in a terminal and see the following

Wolfram Engine activated. See for more information.
Wolfram Language 12.2.0 Engine for Linux x86 (64-bit)
Copyright 1988-2021 Wolfram Research, Inc.

And we can try that it is working

In[1]:= $Version

Out[1]= 12.2.0 for Linux x86 (64-bit) (January 7, 2021)

In[2]:= Integrate[1/(1 + x^2), x]

Out[2]= ArcTan[x]

Now we need to install WolframLanguageForJupyter. For that we can type the following in a terminal

git clone

cd WolframLanguageForJupyter/

./configure-jupyter.wls add

To test that it is installed we can type the following in a terminal

jupyter kernelspec list

and it should have an output that includes a line similar to the following

wolframlanguage12.   /home/nicoguaro/.local/share/jupyter/kernels/wolframlanguage12.2

Or we could also try with

jupyter notebook

and see the following in the kernel menu.

Kernel menu after installing WolframLanguageForJupyter.

Test drive

I tested some of the capabilities and you can download the notebook or see a static version here.

Let's compute the integral

\begin{equation*} \int \frac{1}{1 + x^3}\mathrm{d}x\, . \end{equation*}
sol:= Integrate[1/(1 + x^3), x]
\begin{equation*} -\frac{1}{6} \log \left(x^2-x+1\right)+\frac{1}{3} \log (x+1)+\frac{\tan^{-1}\left(\frac{2 x-1}{\sqrt{3}}\right)}{\sqrt{3}} \end{equation*}

And make a 3D plot.

fun:= Sin[Sqrt[x^2 + y^2]]/Sqrt[x^2 + y^2]
Plot3D[fun, {x, -5*Pi, 5*Pi}, {y, -5*Pi, 5*Pi},
    PlotPoints -> 100, BoxRatios -> {1, 1, 0.2},
    PlotRange -> All]
3D plot in the notebook.

In this case we don't have an interactive image. This is still not implemented, but if you are interested there is an open issue about it in GitHub.

Coming back to the Boundary element method

During October (2017) I wrote a program per day for some well-known numerical methods in both Python and Julia. It was intended to be an exercise to learn some Julia. You can see a summary here. I succeeded in 30 of the challenges, except for the BEM (Boundary Element Method), where I could not figure out what was wrong that day. The original post is here.

Thomas Klimpel found the mistake and wrote an email where he described my mistakes. So, I am creating a new post with a correct implementation of the BEM.

The Boundary Element Method

We want so solve the equation

\begin{equation*} \nabla^2 u = -f(x, y)\quad \forall (x, y) \in \Omega\, , \end{equation*}


\begin{equation*} u(x) = g(x, y), \quad \forall (x, y)\in \partial \Omega \, . \end{equation*}

For this method, we need to use an integral representation of the equation, that, in this case, is

\begin{equation*} u(\boldsymbol{\xi}) = \int_{S} [u(\mathbf{x}) F(\mathbf{x}, \boldsymbol{\xi}) - q(\mathbf{x})G(\mathbf{x}, \boldsymbol{\xi})]\mathrm{d}S(\mathbf{x}) + \int_{V} f(\mathbf{x}) G(\mathbf{x}, \boldsymbol{\xi}) \mathrm{d}V(\mathbf{x}) \end{equation*}


\begin{equation*} G(\mathbf{x}, \boldsymbol{\xi})= -\frac{1}{2\pi}\ln|\mathbf{x} - \xi| \end{equation*}


\begin{equation*} F(\mathbf{x}, \boldsymbol{\xi}) = -\frac{1}{2\pi |\mathbf{x} - \boldsymbol{\xi}|^2} (\mathbf{x} - \boldsymbol{\xi})\cdot\hat{\mathbf{n}} \end{equation*}

Then, we can form a system of equations

\begin{equation*} [G]\{q\} = [F]\{u\}\, , \end{equation*}

that we obtain by discretization of the boundary. If we take constant variables over the discretization, the integral can be computed analytically by

\begin{equation*} G_{nm} = -\frac{1}{2\pi}\left[r \sin\theta\left(\ln|r| - 1\right) + \theta r\cos\theta\right]^{\theta_B, r_B}_{\theta_A, r_A} \end{equation*}


\begin{equation*} F_{nm} = \left[\frac{\theta}{2\pi}\right]^{\theta_B}_{\theta_A} \end{equation*}

for points \(n\) and \(m\) in different elements, and the subindices \(A,B\) refer to the endpoints of the evaluation element. We should be careful evaluating this expression since both \(r_A\) and \(r_B\) can be (close to) zero and make it explode. Also, here it was the main problem since I forgot to compute the angles with respect to elements that are, in general, not aligned with horizontal or vertical axes.

For diagonal terms the integrals evaluate to

\begin{equation*} G_{nn} = -\frac{L}{2\pi}\left(\ln\left\vert\frac{L}{2}\right\vert - 1\right) \end{equation*}


\begin{equation*} F_{nn} = - \frac{1}{2} \end{equation*}

with \(L\) the size of the element.

Following is the code. Keep in mind that this code works for purely Dirichlet problems. For mixed Dirichlet-Neumann the influence matrices would need rearranging to separate known and unknowns in opposite sides of the equation.

You can download the files for this project here. It includes a YML file to create a conda environment with the dependencies listed. For example, it uses version 3.0 of Meshio.

import numpy as np
from numpy import log, arctan2, pi, mean, arctan
from numpy.linalg import norm, solve
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import meshio

  def assem(coords, elems):
      """Assembly matrices for the BEM problem

      coords : ndarray, float
          Coordinates for the nodes.
      elems : ndarray, int
          Connectivity for the elements.

      Gmat : ndarray, float
          Influence matrix for the flow.
      Fmat : ndarray, float
          Influence matrix for primary variable.
      nelems = elems.shape[0]
      Gmat = np.zeros((nelems, nelems))
      Fmat = np.zeros((nelems, nelems))
      for ev_cont, elem1 in enumerate(elems):
          for col_cont, elem2 in enumerate(elems):
              pt_col = mean(coords[elem2], axis=0)
              if ev_cont == col_cont:
                  L = norm(coords[elem1[1]] - coords[elem1[0]])
                  Gmat[ev_cont, ev_cont] = - L/(2*pi)*(log(L/2) - 1)
                  Fmat[ev_cont, ev_cont] = - 0.5
                  Gij, Fij = influence_coeff(elem1, coords, pt_col)
                  Gmat[ev_cont, col_cont] = Gij
                  Fmat[ev_cont, col_cont] = Fij
      return Gmat, Fmat

  def influence_coeff(elem, coords, pt_col):
      """Compute influence coefficients

      elems : ndarray, int
          Connectivity for the elements.
      coords : ndarray, float
          Coordinates for the nodes.
      pt_col : ndarray
          Coordinates of the colocation point.

      G_coeff : float
          Influence coefficient for flows.
      F_coeff : float
          Influence coefficient for primary variable.
      dcos = coords[elem[1]] - coords[elem[0]]
      dcos = dcos / norm(dcos)
      rotmat = np.array([[dcos[1], -dcos[0]],
                      [dcos[0], dcos[1]]])
      r_A =[elem[0]] - pt_col)
      r_B =[elem[1]] - pt_col)
      theta_A = arctan2(r_A[1], r_A[0])
      theta_B = arctan2(r_B[1], r_B[0])
      if norm(r_A) <= 1e-6:
          G_coeff = r_B[1]*(log(norm(r_B)) - 1) + theta_B*r_B[0]
      elif norm(r_B) <= 1e-6:
          G_coeff = -(r_A[1]*(log(norm(r_A)) - 1) + theta_A*r_A[0])
          G_coeff = r_B[1]*(log(norm(r_B)) - 1) + theta_B*r_B[0] -\
                  (r_A[1]*(log(norm(r_A)) - 1) + theta_A*r_A[0])
      F_coeff = theta_B - theta_A
      return -G_coeff/(2*pi), F_coeff/(2*pi)

  def eval_sol(ev_coords, coords, elems, u_boundary, q_boundary):
      """Evaluate the solution in a set of points

      ev_coords : ndarray, float
          Coordinates of the evaluation points.
      coords : ndarray, float
          Coordinates for the nodes.
      elems : ndarray, int
          Connectivity for the elements.
      u_boundary : ndarray, float
          Primary variable in the nodes.
      q_boundary : [type]
          Flows in the nodes.

      solution : ndarray, float
          Solution evaluated in the given points.
      npts = ev_coords.shape[0]
      solution = np.zeros(npts)
      for k in range(npts):
          for ev_cont, elem in enumerate(elems):
              pt_col = ev_coords[k]
              G, F = influence_coeff(elem, coords, pt_col)
              solution[k] += u_boundary[ev_cont]*F - q_boundary[ev_cont]*G
      return solution

#%% Simulation
mesh ="disk.msh")
elems = mesh.cells["line"]
bound_nodes = list(set(elems.flatten()))
coords = mesh.points[bound_nodes, :2]
x, y = coords.T
x_m, y_m = 0.5*(coords[elems[:, 0]] + coords[elems[:, 1]]).T
theta = np.arctan2(y_m, x_m)
u_boundary = 3*np.cos(6*theta)

#%% Assembly
Gmat, Fmat = assem(coords, elems)

#%% Solution
q_boundary = solve(Gmat,

#%% Evaluation
ev_coords =  mesh.points[:, :2]
ev_x, ev_y = ev_coords.T
solution = eval_sol(ev_coords, coords, elems, u_boundary, q_boundary)

#%% Visualization
tris = mesh.cells["triangle"]
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_trisurf(ev_x, ev_y, solution, cmap="RdYlBu", lw=0.3,
plt.savefig("bem_solution.png", bbox_inches="tight", transparent=True,

The result in this case is the following.

Solution of the differential equation using the BEM.

Technical writing: using math

In a previous post I mentioned some general aspects of technical writing. In this one, I would like to talk about including mathematical expressions in technical documents.

There are two main ways to include math in your documents:

  • using text; and

  • using a graphic interface.

Using a graphic interface, such as the equation editor in LibreOffice Writer or MS Word, or MathType is convenient. You don't need to memorize anything and you can look at your expressions while creating them. Nevertheless, it can be slow compared to using text input — once you are comfortable with the syntax.

There are two main flavors of equations used over the internet:

  • MathML is a W3C standard for equations and it is included in HTML5, so it would work in all modern browsers. The problem with it is that it is not designed to be written by hand. So one can use it if have some automatic way of generating the code.

  • LaTeX is my suggested way to write equations. The learning curve might be a little bit steep at the beginning but it pays off.

One tool that helps with equations is MathPix Snip that automatically generates LaTeX or MathML code from an image, even a handwritten one. Another tool that is really useful is Detexify that let you draw a symbol and gives you the LaTeX syntax for it.

In the remaining of the posts I will show my suggestions for working with equations in LibreOffice and MS Word. If you are using LaTeX or MarkDown/ReStructuredText for your documents you are already using LaTeX for your equations.


LibreOffice has its own math editor with its own syntax and it works fine for small expressions, but it gets complicated for large equations or long algebraic manipulations. For LibreOffice I would suggest to use TexMaths, it is simple to use and works for the word processor (Writer) and presentations (Impress). I suppose it also works for spreadsheets (Calc), but I don't remember using equations in one.

MS Office

MS Office has its own math editor as well, it works fine and is easy to use. Nevertheless, the same problem appears when you want long expressions. One option is to directly use LaTeX in Office but I prefer to use IguanaTex. It is a complement that allows to input equations similarly to TexMaths in LibreOffice.

You could also directly paste MathML equations into MS Word (>2013 and Windows).

Use a SymPy

Indepent of the tool that you use to write your documents I strongly suggest to use a CAS (Computer Algebra System), such as Mathematica or SymPy. These programs have automatic generation of LaTeX and MathML from expression and that can ease the process a lot.

Let's check an example. Suppose that we have the function

\begin{equation*} f(x) = \exp(-x^2) \sin(3*x) \end{equation*}

and we want to compute its second derivative

\begin{equation*} f''(x) = \left(- 12 x \cos{\left(3 x \right)} + 2 \left(2 x^{2} - 1\right) \sin{\left(3 x \right)} - 9 \sin{\left(3 x \right)}\right) e^{- x^{2}} \end{equation*}

The following code gives us the LaTex code

from sympy import *
f = exp(-x**2)*sin(3*x)
fxx = diff(f, x, 2)

that is

\left(- 12 x \cos{\left(3 x \right)} + 2 \left(2 x^{2} - 1\right) \sin{\left(3 x \right)} - 9 \sin{\left(3 x \right)}\right) e^{- x^{2}}

That corresponds to the code that I used above to render the equation

If, we wanted the MathML code of that expression we could use the following snippet

from sympy import *
f = exp(-x**2)*sin(3*x)
fxx = diff(f, x, 2)
print(mathml(fxx, printer="presentation"))

notice the extra argument printer="presentation". If we want to add this to MS Word, for example, we could add the output (that I will not show because is really long) inside the following

<math xmlns = "">

When using Jupyter Notebook this can be done graphically with a right click over the expression. Then, the following menu is shown



  1. “How to Insert Equations in Microsoft Word.” WikiHow, Accessed 3 Aug. 2020.

  2. “Copy MathML into Word to Use as Equation.” Stack Overflow, Accessed 3 Aug. 2020.

  3. “Python - Output Sympy Equation to Word Using Mathml.” Stack Overflow, Accessed 3 Aug. 2020.

  4. OERPUB (2016), “Mathconverter”,, Accesed 3 Aug. 2020.

Downloading videos from MS Stream

This week a student asked me about downloading the videos from one of the courses from MS Stream. The problem is that if you are not a proprietary of the video you cannot download it. So, I will show you an option to download videos without being a proprietary of them.

Disclaimer: It might be a good idea if you ask your organization about the copyright of the videos.


You will need the following:

destreamer installation

After getting the prerequisites you can download destreamer using

$ git clone
$ cd destreamer
$ npm install
$ npm run build

in a terminal.

If you do not want to play with environment variables, I suggest that you just add ffmpeg to the same folder as destreamer.


After that, you need to navigate to the folder where you downloaded destreamer and

$ ./ -i ""

in Mac or Linux,

$ destreamer.cmd -i ""

in the command prompt in Windows, and

$ destreamer.ps1 -i ""

in PowerShell.``VIDEO_ID`` refers to the identifier in MS Stream.

If you want to download several files (like a complete course), you can create a file with the URLs and use

$ destreamer.cmd -f filename.txt

Randomized Marking of a Tetrahedron

Yesterday (June 4, 2020), Christian Howard posted on Twitter the following question

You are given a tetrahedron τ. For each triangular facet of τ, we uniformly at random mark one of their edges. What is the probability that there exists an edge of τ that is marked twice?

I thought about a little bit but I couldn't find how to count properly. Then, a number popped up in my mind out of the blue: \(2/3\), but I don't know why. So, I decided to run a simulation to check this number.

The right answer is \(51/81\) that is approximately 63%. This calculation is well explained in Christian's blog and with some cool drawings (and memes).

The algorithm

The algorithm is quite simple. I number the edges in each face following a counter-clockwise orientation:

  • face 0: edge 0, edge 1, edge 2

  • face 1: edge 0, edge 3, edge 4

  • face 2: edge 1, edge 5, edge 3

  • face 3: edge 2, edge 4, edge 5

Then, I take each face and pick a random number from \((0, 1, 2)\) and mark the corresponding edge, and move to the following face. I repeat this process several times and I count the favorable cases and divide them by the number of trials to get an estimate of the probability.

Following is a Python code that follows this idea.

import numpy as np
import matplotlib.pyplot as plt

faces = np.array([
        [0, 1, 2],
        [0, 3, 4],
        [1, 5, 3],
        [2, 4, 5]])

def mark_edges():
    marked_edges = np.zeros((6), dtype=int)
    for face in faces:
        num = np.random.randint(0, 3)
        edge = face[num]
        marked_edges[edge] += 1
    return marked_edges

def comp_probs(N_min=1, N_max=5, ntrials=100):
    prob = []
    N_vals = np.logspace(N_min, N_max, ntrials, dtype=int)
    for N in N_vals:
        cont_marked = 0
        for cont in range(N):
            marked = mark_edges()
            if 2 in marked:
                cont_marked += 1
    return N_vals, prob

#%% Computation
N_min = 1
N_max = 5
ntrials = 100
N_vals, prob = comp_probs(N_min, N_max, ntrials)

#%% Plotting
plt.figure(figsize=(4, 3))
plt.hlines(0.63, 10**N_min, 10**N_max, color="#3f3f3f")
plt.semilogx(N_vals, np.array(prob), "o", alpha=0.5)
plt.xlabel("Number of trials")
plt.ylabel("Estimated probability")
plt.savefig("prob_tet.svg", dpi=300, bbox_inches="tight")

And we can see the following evolution for different number of trials.

Estimated probabilities for different sample sizes.

Technical writing

This is the first post about technical writing * from a series that I will be creating during the course of this year. Technical writing is something that most of us have to deal with in different contexts. For example, in college coursework, research publications, software documentation. The main idea of the series is to mention some of the tricks that I have learned over the years and some tools that might come in handy.

Future posts will (probably) be about:

  • Using equations;

  • Using figures;

  • Using tables; and

  • Managing bibliographic references.

The current post

As mentioned above, technical writing is something that a lot of persons have to deal with. This is a skill that is sometimes overlooked, but it should not. According to the U.S. Bureau of Labor Statistics

Technical writers prepare instruction manuals, how-to guides, journal articles, and other supporting documents to communicate complex and technical information more easily.

And it is a desired skill in the workplace. Its demand is expected to grow around 10% in the current decade.


The first thing that I should mention is that writing documents is typography. "Putting documents" together is typography because we are designing with text (Butterick, 2019). So, we should consider ourselves typographers since we are constantly designing documents.

I would suggest taking a look at "Butterick's Practical Typography" since it is a really good book about it and it reads smoothly. I will mention some important points here according to Butterick's "Typography in ten minutes":

  1. The most important typographic selection is on the body text. This is due to the fact that it is most of the document.

  2. Choose a point size between 10-12 points for printed documents and 15-25 pixels for digital documents.

  3. Line spacing should be between 120-145 % of the point size.

  4. Line length should be between 45-90 characters. This is roughly 2 or 3 small caps alphabets:


  5. Mind the selection of your font. Try to avoid default fonts such as Arial, Calibri or Times New Roman.


Another point that I want to touch in this post is about editors. The first question that arises is "what editor should I use?". The short answer is: use whatever your peers are using. That's my best advice; that way you have people to discuss with you about your doubts.

The long answer … is that each editor has its weak and strong points. I have written scientific papers in LaTeX, LibreOffice Writer and MS Word; all of them look professional. So, in the end, you can write your documents in several ways and achieve a similar result. I prefer to use LaTeX for long documents since it is centered in the structure of the document instead of the appearance and this is the way one should manage a long document like a dissertation, in my opinion.

If you just want me to pick one editor and suggest it to you, I would say that you should ride with LibreOffice. A good reference for it is "Designing with LibreOffice". Once you learn how to use styles you will ask how have you been writing documents all this time.

There are two main flavors for editors that I am going to discuss: WYSIWYG (What You See Is What You Get) and markup-based editors.

  • WYSIWYG. This category is the one that most people is familiar with. Two examples are:

    • LibreOffice writer; and

    • Microsoft Word.

  • Markup-based editors rely on marks on the "text" to differentiate sections and styles. In this case, your text looks like code, as seen in the following image


    Some examples are:

Independently of what your main editor is I suggest that you use Pandoc. It allows you to convert between several formats, making the process a little bit easier. There is even an editor based completely on it named Panwriter.


  1. Matthew Butterick (2019). Butterick's Practical Typography. Second edition, Matthew Butterick.

  2. Wikibooks contributors. (2020). LaTeX. Wikibooks, The Free Textbook Project.

  3. Bruce Byfield (2016). Designing with LibreOffice. Friends of OpenDocument, Inc.

  4. Deville, S. (2015). Writing academic papers in plain text with Markdown and Jupyter notebook. Sylvain Deville.

  5. Eric Holscher (2016). An introduction to Sphinx and Read the Docs for Technical Writers


This post is (somewhat) related to a previous post where I discussed research tools that most of us need but are not commonly taught in a formal fashion.

Spell Check in Jupyter Notebook

The purpose of this post is to show how to have automatic spell check in Jupyter Notebook, as shown below.

Example of spell checking in Jupyter Notebook.

There are several ways to do this. However, the easiest way is through the (nbextension) Spellchecker. plugin.

Step by step

The steps to follow are those:

  1. Install Jupyter notebook extensions (nbextensions). This includes Spellchecker.
  2. Locate the dictionaries in the folder where the plugin is. Dictionaries must use UTF-8 encoding.
  3. Configure the path of the dictionaries. This can be a URL or a path relative to the folder where the plugin is located.

We will describe each step in detail below.

Step 1: Install nbextensions

There is a list of plugins that add some commonly used functionality to the Jupyter notebook.

Type the following in a terminal, to install it using PIP.

pip install jupyter_contrib_nbextensions

However, if Anaconda is being used the recommended method is to use conda, as shown below.

conda install -c conda-forge jupyter_contrib_nbextensions

This should install the plugins and also the configuration interface. In the main menu of Jupyter notebook a new tab named Nbextension will appear. Here you can choose the add-ons to use. The appearance is as follows.

Graphical interface for Jupyter plugins.

Some recommended plugins are:

  • Collapsible Headings: that allows to hide sections of the documents.
  • RISE: that turns notebooks into presentations.

Step 2: Dictionaries for Spanish

The documentation of Spellchecker suggests using a Python script to download dictionaries from project Chromium. However, these are encoded in ISO-8859-1 (western) and it fails for characters with accents or tildes. So, to avoid problems the dictionary must be UTF-8. They can be downloaded at this link.

Once you have the dictionaries, they must be located in the path of the plugin. On my computer this would be


and within this we will place them in


This location is arbitrary, the important thing is that we need to know the relative path to the plugin.

Step 3: Plugin Configuration

Now, in the Nbextensions tab we select the plugin and fill the fields with the information from our dictionary:

  • language code to use with typo.js: es_ES
  • url for the dictionary .dic file to use: ./typo/dictionaries/es_ES.dic
  • url for the dictionary .aff file to use: ./typo/dictionaries/es_ES.aff

This is shown below.

Configuration with local files.

Another option is to use the URL for the files. The dictionaries of the project hunspell in UTF-8 are available at In this case, the configuration would be:

  • language code to use with typo.js: es_ES
  • url for the dictionary .dic file to use:
  • url for the dictionary .aff file to use:

And it is shown below.

Configuration with remote files.

#SWDchallenge: Graph makeover

At storytelling with data, they run a monthly challenge on data visualization named #SWDchallenge. The main idea of the challenges is to test data visualization and storytelling skills.

Each month the challenge has a different topic. This month the challenge was to improve an existing graph. I decided to take a previous graph of mine that was published on a paper that was published on 2015.

The "original"

The following is the graph that we are going to start with.

Original plot

This is not the exact same graph that was presented in the article on 2015, but it serves the purpose of the challenge. The data can be downloaded from here. This graph present the fraction of energy (\(\eta\)) transmitted for helical composites with different geometrical parameters. The parameters varied were:

  • Pitch angle \(\alpha\): the angle between consecutive layers;

  • Composite thickness \(D\), that is normalized to the wavelength \(\lambda\); and

  • Number of layers \(N\) in each cell of the composite.

The following schematic illustrate these parameters.


I would not say that the graph is awful, and, in comparison to what you would find in most scientific literature it is even good. But … in the land of the blind, the one-eyed is king. So, let's enumerate what are the sins of the plot and see if we can correct them:

  • It has two x axes.

  • The legend is enclosed in a box that seems unnecessary.

  • Right and top spines are not contributing to the plot.

  • Annotations are stealing protagonism from the data.

  • It looks clutterd with lines and markers.

  • It is a spaghetti graph.

The following snippet generates this graph.

import numpy as np
from matplotlib import pyplot as plt
from matplotlib import rcParams

rcParams[''] = 'serif'
rcParams['font.size'] = 16
rcParams['legend.fontsize'] = 15
rcParams['mathtext.fontset'] = 'cm'

markers = ['o', '^', 'v', 's', '<', '>', 'd', '*', 'x', 'D', '+', 'H']
data = np.loadtxt("Energy_vs_D.csv", skiprows=1, delimiter=",")
labels = np.loadtxt("Energy_vs_D.csv", skiprows=0, delimiter=",",
                    usecols=range(1, 9))
labels = labels[0, :]

fig = plt.figure()
ax = plt.subplot(111)
for cont in range(8):
    plt.plot(data[:, 0], data[:, cont + 1], marker=markers[cont],

# First x-axis
xticks, xlabels = plt.xticks()
plt.xlabel(r"Number of layers - $N$", size=15)
plt.ylabel(r"Fraction of Energy - $\eta$", size=15)
ax.legend(loc='center left', bbox_to_anchor=(1, 0.5))

# Second x-axis
ax2 = ax.twiny()
plt.xlabel(r"Angle - $\alpha\ (^\circ)$", size=15)

plt.savefig("energy_vs_D_orig.png", dpi=300)


I will vindicate the graph one sin at a time, let's see how it turns out.

It has two x axes

I, originally, added two axes to show both the number of layers and the angle between them at the same time. The general recommendation is to avoid this, so let's get rid of the top one.

First iteration

Legend in a box

Pretty straightforward …

Second iteration

Right and top spines

Let's remove them

Third iteration

Annotations are stealing protagonism

Let's move all the annotations to the background by changing the color to a lighter gray.

Fourth iteration

Clutterd with lines and markers

Let's just keep the lines.

Fifth iteration

And increase the linewidth.

Sixth iteration

It is a spaghetti graph

I think that a good option for this graph would be to highlight one line at a time. Doing this, we end up with.

Seventh iteration

The following snippet generates this version.

import numpy as np
from matplotlib import pyplot as plt
from matplotlib import rcParams

# Plots setup
gray = '#757575'
plt.rcParams["mathtext.fontset"] = "cm"
plt.rcParams["text.color"] = gray
plt.rcParams["xtick.color"] = gray
plt.rcParams["ytick.color"] = gray
plt.rcParams["axes.labelcolor"] = gray
plt.rcParams["axes.edgecolor"] = gray
plt.rcParams["axes.spines.right"] = False
plt.rcParams[""] = False
rcParams[''] = 'serif'
rcParams['mathtext.fontset'] = 'cm'

def line_plots(data, highlight_line, title):
    for cont, datum in enumerate(data[:, 1:].T):
        if cont == highlight_line:
            plt.plot(data[:, 0], datum, zorder=3, color="#984ea3",
            plt.plot(data[:, 0], datum, zorder=2, color=gray,
                     linewidth=1, alpha=0.5)

data = np.loadtxt("Energy_vs_D.csv", skiprows=1, delimiter=",")
labels = np.loadtxt("Energy_vs_D.csv", skiprows=0, delimiter=",",
                    usecols=range(1, 9))
labels = labels[0, :]

plt.figure(figsize=(8, 4))
for cont in range(8):
    ax = plt.subplot(2, 4, cont + 1)
    title = r"$D/\lambda={:.3g}$".format(labels[cont])
    line_plots(data, cont, title)
    plt.ylim(0.4, 1.0)
    if cont < 4:
        plt.xlabel(r"Number of layers - $N$")
    if cont % 4 > 0:
        plt.ylabel(r"Fraction of Energy - $\eta$")

plt.savefig("energy_vs_D_7.png", dpi=300)

Final tweaks

Using Inkscape I added some final details to get the following version.


Further reading

Isometric graphics in Inkscape: Part 2

Last week I posted a quick guide on isometric drawing using Inkscape. In that post, I showed how to obtain images that are projected to the faces of an isometric box.

After my post, I was asked by Biswajit Banerjee on Twitter if I could repeat the process with a more complex example, and he suggested the following schematic

Computing the moment of force in a beam

which, I guess, was created in Inkscape using the "Create 3D Box" option.

In this post, I will:

  1. Use the same approach from last week to recreate this schematic

  2. Suggest my preferred approach for drawing this schematic

First approach

I will repeat the cheatsheet from last week. Keep in mind that Inkscape treats clockwise rotation as positive. Opposite to common notation in mathematics.

Cheatsheet for isometric schematics in Inkscape

Then, to create a box with desired dimensions we first create each rectangle with the right dimensiones (in parallel projections). In the following example we used faces with aspect ratios 3:2, 2:1 and 4:3. The box at the right is the figure obtained after applying the transformations described in the previous schematic.

Orthogonal views and final isometric figure

We can now proceed, to recreate the desired figure. I don't know the exact dimensions of the box; my guess is that the cross-section was a square and the aspect ratio was 5:1. After applying the transformations to each rectangle we obtain the following (adding some tweaks here and there).

Recreated figure using the described approach

Second approach

For this type of schematic, I would prefer to create an axonometric grid (File > Document Properties > Grids). After adding the grid to our canvas it is pretty straightforward to draw the figures in isometric view. The canvas should look similar to the following image.

Second approach: using an axonometric grid

We can then draw each face using the grid. If we want to be more precise we can activate Snap to Cusp Nodes. The following animation shows the step by step construction.

Step by step construction of the isometric

And we obtain the final image.

Recreated figure using the second approach


As I mentioned, Inkscape can be used for drawing simple figures in isometric projection. Nevertheless, I strongly suggest to use a CAD like FreeCAD for more complicated geometries.

Isometric graphics in Inkscape

Sometimes I find myself in need of making a schematic using an isometric projection. When the schematic is complicated the best shot is to use some CAD like FreeCAD, but sometimes it's just needed in simple diagrams. Another situation where this is a common needed is in video games, where "isometric art" and pixel art are pretty common.

What we want is depicted in the following figure.

Views names in third angle representation

That is, we want to start with some information that is drawn, or written in the case of the example, and we want to obtain how would it been seen on one of the faces of an isometric box.

Following, I will describe briefly the transformations involved in this process. If you are just interested in the recipe for doing this in Inkscape, skip to the end of this post.

Transformations involved

Since we are working on a computer screen, we are talking of 2 dimensions. Hence, all transformations are represented by 2×2 matrices. In the particular example of interest in this post we need the following transformations:

  1. Vertical scaling

  2. Horizontal skew

  3. Rotation

Following are the transformation matrices.

Scaling in the vertical direction

The matrix is given by

\begin{equation*} M_\text{scaling} = \begin{bmatrix} 1 &0\\ 0 &a\end{bmatrix}\, , \end{equation*}

where \(a\) is the scaling factor.

Horizontal skew

The matrix is given by

\begin{equation*} M_\text{skew} = \begin{bmatrix} 1 &\tan a\\ 0 &1\end{bmatrix}\, , \end{equation*}

where \(a\) is the skewing angle.


The matrix is given by

\begin{equation*} M_\text{rotation} = \begin{bmatrix} \cos a &-\sin a\\ \sin a &\cos a\end{bmatrix}\, , \end{equation*}

where \(a\) is the rotation angle.

Complete transformation

The complete transformation is given by

\begin{equation*} M_\text{complete} = M_\text{rotation} M_\text{skew} M_\text{scaling}\, , \end{equation*}


\begin{align*} &M_\text{side} = \frac{1}{2}\begin{bmatrix} \sqrt{3} & 0\\ -1 &2\end{bmatrix}\approx \begin{bmatrix} 0.866 & 0.000\\ -0.500 &1.000\end{bmatrix}\, , \\ &M_\text{front} = \frac{1}{2}\begin{bmatrix} \sqrt{3} & 0\\ 1 &2\end{bmatrix}\approx \begin{bmatrix} 0.866 & 0.000\\ 0.500 &1.000\end{bmatrix}\, , \\ &M_\text{plant} = \frac{1}{2}\begin{bmatrix} \sqrt{3} & -\sqrt{3}\\ -1 &1\end{bmatrix}\approx \begin{bmatrix} 0.866 & -0.866\\ 0.500 &0.500\end{bmatrix}\, . \end{align*}

The values seem a bit arbitrary, but they can be obtained from the isometric projection itself. But that explanation would be a bit too long for this post.

Tranformation in Inkscape

We already have the transformation matrices and we can use them in Inkscape. But first, we need to understand how it works. Inkscape uses SVG, the web standard for vector graphics. Transformations in SVG are done using the following matrix

\begin{equation*} \begin{bmatrix} a &c &e\\ b &d &f\\ 0 &0 &1\end{bmatrix}\, , \end{equation*}

that uses homogeneous coordinates. So, one can just press Shift + Ctrl + M and type the components of the matrices obtained above for \(a\), \(b\), \(c\), and \(d\); leaving \(e\) and \(f\) as zero.

My preferred method, though, is to apply each transformation after the other in the Transform dialog (Shift + Ctrl + M). And, this is the method presented in the cheatsheet at the bottom of this post.


If you are just interested in the transformations needed in Inkscape you can check the cheatsheet below. It uses third-angle as presented below.

Views names in third angle representation


Keep in mind that Inkscape treats clockwise rotation as positive. Opposite to common notation in mathematics.

Cheatsheet for isometric schematics in Inkscape