How were you taught science? From textbooks? If yes, then you've obviously walked down the much less interesting path toward understanding science's full power, beauty, and ubiquity.

Science is not the same as it was before. In the time of Planck, Heisenberg, Maxwell, Einstein, etc., it was reserved for the blackboards, for the esoteric labs of Princeton and Zurich, for high-profile conferences that were hoped to change the face of human understanding as well as, disturbingly, world war. The layman’s access to it was limited if not non-existent.

Today, two changes are visible since those times. One: Science is much more accessible in the form of freely available tools, labs whose equipment you can hire, and blogs and courses online to learn whatever you like. Two: As the understanding of our reality became finer, science’s role in policy-making has become stronger and mandatory.

With the onslaught of information if only you opened your cognitive gateways, there is a scientific element in every bit of information we come across. Even biases, promulgated inadvertently by the shortcomings of the social network, are the product of imperfect technology built with scientific principles in mind. The entire media landscape, which rests on the shoulders of information and communication technology, only presents a stroll through years of rational thought.

In this “new-science” world, understanding what science is involves the addition of a new layer of meaning to our lives. The realm of scientific thought, of logic and algebra and geometry, must needs no longer exist as 2D shapes in our textbooks but as a pump of new ideas, inspiration for new solutions, present wherever we choose to look.

In other words, we need contemporary tools to participate in the contemporary use of science. As web-developer Bret Victor put it,

"The power to understand and predict the quantities of the world should not be restricted to those with a freakish knack for manipulating abstract symbols."

To this end: Here are some visualisations that can bring mathematics and physics to life for you.

Fractal Zoomer

A fractal is a shape or a function that exhibits internal symmetry. This means the ratios of various lengths and breadths of the shape’s insides are simply a scaled-down version of the ratios making up the larger shape itself.

Consider a snowflake: it’s intricate fractal design is unique in nature, and can easily set you thinking where nature gets such ideas from. Actually, it’s not that hard.

Start with an equilateral triangle. With each of its three sides as a base, draw a new triangle. Now, you’ll have a shape with 12 distinct sides. With each of those sides as a base, draw a new triangle. After six or seven iterations of this, you’ll end up with what’s called a Koch snowflake, and it looks like this.


Image: Wikimedia Commons

Now, Paul Neave’s Fractal Zoomer demonstrates a much more complex fractal, called a Mandelbrot set. Approximately, it comprises a large bulb, at multiple points on whose circumference are branches of smaller bulbs which tail off in the pattern of the larger bulb itself. Here’s a look.

As you keep zooming in, you see small-scale repititions of the larger structure!

Mathematica & Mathics

Have you ever played around with the Wolfram Mathematica? If you haven’t, please, please do! Download a trial version and experience the simplicity of working with mathematics. With simple commands - in a language mostly like written English - see how numbers and functions work with colours, surfaces, graphs, engineering systems, and so on, all on your desktop.

Mathics is simply Mathematica that’s been ported to Python, and even runs in your browser. It’s lighter, and is open to customisation as well.

Yavkata: Bezier curve

Along with the Bessel function, the Bezier curve is one of the most aesthetically pleasing graph-plots to me. Fortuitously, the Bezier curve is defined in such a way that it is aesthetic.

Unlike most conventional curves, which are plotted by joining the dots of some function, for the Bezier, at least two control points are defined, and then the curve is plotted such that it varies along those control points.

Such a curve has many applications, most importantly in modeling. Whether you’re using Photoshop or AutoCAD, creating an aesthetically pleasing design is an important goal; it doesn’t matter if you’re drawing a cartoon or a car (but it matters if you’re designing a bucket-wheel excavator because at those scales, you don’t care for design!).

In such a case, knowledge of the existence of the Bezier curve should make things easy for you. Put in two points - these are the start- and end-points of the curve. Next, drop control points where you think appropriate. Remember how you move a tangent around in Photoshop to adjust how much a curve bends at a point? You're only repositioning the control points.

For reference, join the start- and end-points in a straight line. So, if there are an equal number of control points above and below this line, the Bezier curve will vary in equal measure above and below this line. Crowding a region with many control points pulls the Bezier toward that point.

For an arbitrarily defined Bezier curve with six control points, this is what you should see:


Image: Wikimedia Commons

Unfortunately, Wiki doesn’t allow you to modify the positions of the control points and let you see how the curve changes. Fortunately, Yavor Atanasov does. His Actionscript class (code available here) lets you play around with the plot, eventually letting you pull a slider that animates your Bezier even as it is plotted.

Kill Math

Kill Math is a wide-ranging project undertaken by Bret Victor, the web-developer, to build applications that let users engage with advanced mathematics in an intuitive, operations-guided manner. For instance, his Scrubbing Calculator lets you play around with algebraic equations and functions without any variables. Instead, you get to manipulate numbers themselves and their relationships with each other to see what you're creating!

My favourite Kill Math tool is the interactive exploration of a dynamical system. Victor's described it using the Lotka-Volterra equations, but you don't actually encounter them. These equations give the rise and fall of predator and prey populations with respect to each other (if the natural resources are the same over a period of time). My own model of them - "solved" using Wolfram Mathematica - look like this.

Victor's model looks like this.

Portal

This is a biggie! Have you played Portal or its successor Portal 2? In the game you're pitted against an artificially intelligent computer that has control of a research facility. Your weapon is a special gun born out of the facility that lets you shoot 'open' and 'close' portals such that if you step into the 'open' one, you emerge out of the 'close' one. Your task is to complete all of the labyrinthine tasks the AI puts you through and then take on the AI itself.


A screencapture of a game of Portal in progress

The best part about the game, though, is how you use the laws of physics in different ways to guide you to your goals! For instance, in many parts of the game, you're asked to cross a wide gap containing an acidic substance by jumping over it. If you try, you'll see that simply running up to the gap and jumping won't do. Instead, you spot a ledge nearby and get atop it. Next, you fire a vertical 'close' portal behind you and a horizontal 'open' portal on the step lower than the one you're on.


Image: Wikimedia Commons

Just jumping would've taken you from zero initial speed to some higher speed, but what about coming into the jump with a more-than-zero initial speed? That's what you're doing. You jump high and into the portal below, and emerge from the one behind you already moving at a fair, non-zero speed, one that's enough to get you safely across the gap.

What have you learnt? The law of conservation of linear momentum, that's what.

Because it's an FPS exercise, the requirements of you and your surroundings are more real than they'd be if they were being read out to you from a textbook. While the threat of dying isn't real enough, you'll also be involved in the game, and dying at that point wouldn't be something you'd lile (especially if you know how annoying the AI is!).

World of Goo

Here's another game, and one not so well-known as Portal is. In the world of Goo, you build bridges, towers, levels, cranes and all other kinds of mechanisms by joining together a bunch of lively black, tar-like balls so they can get from one place to another. The forces of nature are at work here, too, like they're in Portal. The game also teaches you economics by limiting the number of goo-balls at each level.


Screen-captures of World of Goo at various stages

I found this game in my third year of college, studying mechanical engineering, and its aid was fantastic for my materials science course. Of course, I'm not manipulating different kinds of goo, but I did learn how stresses and strains affected structures, how they arose, and how they could be resolved with minimum material.

Attractors: Lorenz butterfly

Science harbours a lot of metaphysical concepts that aren't abstract. One of them is the attractor. An attractor is something to which something else aspires to become... mathematically.

For example, if one number describes the frequency with which a planet revolves around the Sun and another number describes the frequency with which the distance between the planet and the Sun oscillates, and if one of these numbers is not a rational factor/multiple of the other, then the path of the planet around the Sun aspires, over many revolutions, to describe a limit-torus (as depicted below).


Image: Wikimedia Commons

There are different kinds of attractors for different situations. A famous one is the Lorenz butterfly: It describes how two systems that start with very similar initial conditions could end up with completely different end-states because of a phenomenon called sensitive dependence on initial conditions.

Note that the attractors themselves aren't the examples; instead, seeing how the attractor itself behaves so deviantly, you can imagine how a system aspiring to behave like the attractor would!

More colloquially, the practical analogy of this is the butterfly effect - which probably gets its name more from the shape of the Lorenz butterfly curve than from the example of a butterfly flapping its wings to kick-start a tornado faraway. Anyway, here's a web-based simulator of Lorenz butterfly curves different initial starting points.

Tsunami research: NOAA Center for Tsunami Research

This is a simple example in that its creation didn't require as much clever programming as the previous examples did. The idea is to present as much relevant detail as possible that goes with studying the seas in which quake-triggered tsunamis are likely to occur. The base layer is a Google Maps map on which NOAA data has been superimposed.


Image: NOAA

Clicking on the links in the cards that appear when you hover over different locations in the map takes you to a different page (unfortunately - instead of opening in a smaller hovering box next to your mouse) where you will be shown tabulated numbers.

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That's all I got... for now. But I'm sure that if you went looking by yourself, you'd find a lot more, and probably better examples than I have. These are simply the ones I regularly have used or use for inspiration. In fact, for the more programmatically inclined, don't forget to check out d3.js (where I first learned about Voronoi diagrams and clustering algorithms).

Looking at each one of them, I can't help but think that each scientific concept like the Bezier curve or the fractal is, in turn, composed of more scientific concepts, especially ones of geometry and scaling. Just like that, working with them has taken me, and will take you, on a long and exciting journey of exploration and discovery.