Week 5: Mathematics Visualised / by Valzorra

Visualising Probability

We’ve already discussed Chance and Probability in great detail in Week 1: A Welcome Return, Week 2: Calculus and Cancer, Week 3: Research on Mathematics, Week 3: The Matrix, Week 4: Markov Chains, and Week 4: Probability Manipulation in SEO. Although all of that study has given great understanding of how one can control and manipulate probability to make educated and diverse design choices, I haven’t really discussed how Probability is visualised in Mathematics (aside from the Probability Trees discussed with James). In order to visualise probability mathematicians and algorithmic artists turn to Probability Distributions.

The foundation of all Probability Distribution is the Uniform Distribution. In Uniform Distribution there is an equal chance of achieving each result from a certain set or range. For example, with Fair Dice (for more information go to Week 3: Research on Mathematics) we would use Uniform Distribution to visualise the chance of landing on each number of the dice as by definition the chances are equal. Uniform Distribution is also the default for nearly every random() function in programming. It’s interesting to note that when although Uniform Distribution attempts to follow true randomness, it does then to form small clusters and clumps in certain areas. Nonetheless, this Probability Distribution is fantastic for achieving an equal coverage of an area and would be visualised in one dimensional and two dimensional space as illustrated below.

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Gaussian or Normal Distribution is extremely popular both in nature and in everyday human activity. In Gaussian Distribution, commonly referred to as a Bell Curve, the data tends to be focused around the centre values with no bias towards the left or the right. In this Probability Distribution, values closer to the mean (which is also equal to the median and the mode) tend to be chosen more often. What’s noteworthy about Normal Distribution is the idea of Standard Deviation, which simply indicates how spread out the values are. Typically, 68% of all values fall within one Standard Deviation of the mean on either side, 95% of values fall within two Standard Deviations from the mean, while 99.7% fall within three. Visually, Normal Distribution tends to focus around the centre, with few values spread out on the sides as shown below.

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Another way of visualising probability is through the Power Law Distribution, the most popular of which is the Pareto Distribution. Power Law Distribution is exponentially skewed towards the minimum possible value. These type of Distributions are most commonly used to describe observable phenomena. The most notable example that uses this format is for the Distribution of Wealth in Society as that illustrates how a small portion of the population holds substantial amounts of wealth. Aesthetically, this distribution achieves a good balance between different sized objects. Pareto Distribution in both one dimensional and two dimensional space is shown below.

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Probability in Generative Art

The Probability Distributions mentioned above are significant to the way we represent and structure data and to the study of Data Visualisation. However, there are also a number of artworks that take advantage of their properties and logic, specifically Algorithmic Art and Generative Art. Both of those art fields take advantage of autonomous systems such as machines, computers, language rules, genetic sequences, algorithms, and more. Due to the nature of the autonomous systems, the artworks are often created with an element of chance and probability, which can be controlled by the artist or designer to achieve certain effects. The idea of controlling chance and probability through algorithms and mathematics strongly ties into what interests me about the field, which is why Generative Art is perfect visualisation of that concept. Below I have provided a few notable examples of Generative and Algorithmic Art by Tyler Hobbs, which illustrate the field’s relationship to order and randomness, to aesthetics and mathematics. The works below take advantage of Gaussian Distribution and Power Law Distribution in order to obtain a good balance of shapes and density within the works, making them appear clean, precise, yet packed with information.

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Data Visualisation

Data Visualisation is the study of the representation of data in the most clear and easy to communicate manner possible. Oftentimes it involves taking vast amounts of data, involving complex and intertwining relationships, and representing it in a visual and digestible format. While Matrices and Probability allow us to manipulate data, Data Visualisation allows the untrained eye to work with that information and understand the contents of a Matrix, for example. There are dozens of different and innovative ways to represent data, both in two-dimensional and three-dimensional space, including Arc Diagrams, Density Plots, Chord Diagrams and many more (for more examples, have a look at Week 2: The Four Themes). Data Visualisation incorporates elements of Graphic Design, User Experience Design, User Interface Design, Mathematics, the Sciences, Art and is all round an incredibly useful discipline for visual communication. Below I have attached a few notable examples and creative approaches to Data Visualisation, which could also be referred to as Data Art. What I find incredibly exciting about Data Visualisation is the idea of clear and concise communication of complex ideas through the power of good design. This is a method of helping people understand information, it can be incredibly educational, and can help train people who have no prior knowledge of the subject matter. Therefore, Data Visualisation is a field worth investigation for both designers and educators.

In the project below, Datamatrics, Ryoji Ikeda has taken enormous volumes of data from computer generated code from software and hardware errors and bugs. All of these errors have then been represented in a variety of two-dimensional and three-dimensional formats. The project is not a traditional use of Data Visualisation as it is actually aimed at representing what can often be the chaotic nature of code and computer software. There has been a later iteration of this specific project, which explored the concept in purely binary values and further reiterated the idea of simplicity and chaos within the realms of technology.

The work below by Marcin Ignac represents the Global Sea Transportation Network is entirely based on the paths of sea transportation vessels and their actual GPS Coordinates as they travel around the world. This project is especially connected to Matrices as the data from those GPS Coordinates would be recorded within a Matrix. Therefore, this is a terrific example of how Matrices can be presented in a very easy to understand and digestible format, allowing for more clear communication of the data. What’s more is that by adding new routes and new matrices within this system, it would be incredibly easy to observe changes and estimate where certain ships are meant to be at any given time.

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Platonic Solids Transformations

I have gone into detail over the Platonic Solids and why they are exciting in some of my previous posts (Week 3: Research on Geometry, Week 3: Research on Mathematics). However, what’s even more interesting is that they can actually be used as the basis for a near infinite number of other complicated shapes through the use of Geometric Transformations. I found a fantastic tool online created by Marcin Ignac, which can edit Platonic Solids to create a multitude of more complex shapes. Some of the techniques used in the tool include splitting the faces of the Platonic Solids into triangles, changing the heights of those triangles. Additionally, shapes can be merged together, their faces and geometric properties working in combination with each other to create a new solid. Overall, the tool is an absolutely fantastic demonstration of Transformations within three-dimensional space and what’s more is that any figure can also be exported as an OBJ. There transformations can be turned into separate mechanics and functions within a game, making the tool and the mathematics behind it a method for idea generation. I highly recommend exploring the Platonic Solids Editor as it provides valuable insight into many of the geometric properties discussed throughout the blog.