The 4th dimension in our case is where the 3D structures including this very Universe combine and exist within changing time frames. 4D structures can’t exist within 3D ones but 3D structures can exist in a 4D just like your drawings exist within that flat paper as lines and points but couldn’t exist in our 3D world by itself. Extra dimensions work the same, like a Matryoshka doll that loses and or gains properties the further you go.
Image: 3D projection of a tesseract undergoing a simple rotation in four dimensional space.
In mathematical physics, Minkowski space or Minkowski spacetime (named after the mathematician Hermann Minkowski) is the mathematical space setting in which Einstein’s theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime. [**]
In physics, spacetime (also space–time, space time or space–time continuum) is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as existing in three dimensions and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions. From a Euclidean space perspective, the universe has three dimensions of space and one of time. By combining space and time into a single manifold, physicists have significantly simplified a large number of physical theories, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels. [**]
But my favorite explanation of extra dimensions in general is Carl Sagan’s version. His version was based on Flatland: A Romance of Many Dimensions which is an 1884 satirical short story by Edwin Abbott Abbott:
The story is about a two-dimensional world referred to as Flatland which is occupied by geometric figures. Women are simple line-segments, while men are polygons with various numbers of sides. The narrator is a humble square, a member of the social caste of gentlemen and professionals in a society of geometric figures, who guides us through some of the implications of life in two dimensions. The Square has a dream about a visit to a one-dimensional world (Lineland) which is inhabited by “lustrous points”.
He attempts to convince the realm’s ignorant monarch of a second dimension but finds that it is essentially impossible to make him see outside of his eternally straight line.
He is then visited by a three-dimensional sphere, which he cannot comprehend until he sees Spaceland for himself. This Sphere (who remains nameless, like all characters in the novella) visits Flatland at the turn of each millennium to introduce a new apostle to the idea of a third dimension in the hopes of eventually educating the population of Flatland of the existence of Spaceland. From the safety of Spaceland, they are able to observe the leaders of Flatland secretly acknowledging the existence of the sphere and prescribing the silencing of anyone found preaching the truth of Spaceland and the third dimension. After this proclamation is made, many witnesses are massacred or imprisoned (according to caste).
After the Square’s mind is opened to new dimensions, he tries to convince the Sphere of the theoretical possibility of the existence of a fourth (and fifth, and sixth …) spatial dimension.
The depiction above is a 4 dimensional figure as represented by 3 dimensional cubes within cubes to visualize how 4th dimensions may work.
Related: Carl Sagan explains extra dimensions
Two bubbles merge in an arrangement which minimises the total surface area, given the air stored in each compartment is equal to the original air in each bubble. [A single bubble is a sphere, which is the minimum surface are for a given volume.] The two compartments are parts of spheres, and the boundary between them is part of another sphere meeting the other walls at 120 degree angles. If the two bubbles were originally the same size, the boundary sphere has an infinite radius, giving a flat wall. Though it is a familiar picture to anyone who has blown bubbles, it was only proven that this was how double bubbles are made in 2002. [more1] [more2] [code]
Strange Attractors by Chaotic Atmospheres
The darkest art known as Chaos Theory is perfectly embodied in the form of its strange attractors: vast looping trajectories of variables that, when plotted, conjure gorgeous yet insidiously disruptive patterns. Chaotic Atmosphere’s Math: Rules series pays tribute to the beautiful form of chaos and its inevitable collapse of all our efforts to predict it.
I want them ALL.
Metal-loving researchers analyzed the collective movement of individuals in mosh pits, which could help explain mass movements in other extreme situations.
The algorithm that won an Oscar
Hollywood likes a good explosion. Now, with the help of an open source algorithm called Wavelet Turbulence, filmmakers can digitally create pyrotechnics that were formerly time-consuming and difficult to control.
UCSB’s Theodore Kim (along with three collaborators) picked up the Academy Award in Technical Achievement for Wavelet Turbulence. The algorithm uses a theory of turbulence developed in the 1940s by Russian mathematician Andrey Kolmogorov.
So far, it has been used in over 26 major hollywood productions including Avatar, Sherlock Holmes, Hugo, and Super 8 (pictured above).
So What Is a Fractal, Anyway?
It is my firm conviction that mathematics underpins everything in the universe. Mathematics was intrinsically designed to explain the world; Newton invented differential and integral calculus, for example, to explain why the orbits of the planets were in the shape of an ellipse. Understanding the mathematics behind phenomena will not make them any less beautiful; it will make them more beautiful in that the complexity of our world is better understood and appreciated.
To put it simply, fractals are self-similar and have an unusual relationship with space - kind of like the ‘It’s Complicated’ relationship status on Facebook. Self-similar, in this case, means that fractals have something called irreducible complexity: No matter how far you “zoom in” to a fractal pattern, the structure will be exactly or approximately similar to the fractal itself. In order to understand a fractal’s unusual relationship with space, however, we’re going to need to consider the concept of dimensions.
Dimensions, in their simplicity, are a very interesting mathematical concept. What, for example, makes a line one-dimensional, and a plane two-dimensional? Note that lines and planes are both self-similar, as outlined by the definition above. The differences actually lie in their self-similarity; a line can be broken down into N self-similar pieces, each with a magnification factor of N; a plane can be broken down into N^2 similar pieces, each with a magnification factor N; and a cube into N^3 pieces, with the same magnification factor N. A magnification factor will be defined here as the amount you have to “zoom in” to regain the original structure.
Therefore, dimensions of self-similar objects are simply the exponent of the number of self-similar pieces with magnification factor N into which the figure may be broken. So the line segment mentioned before exists in one dimension; the plane in two dimensions; and the cube in three dimensions. Not too bad, right?
Well, here’s the complicated part. Fractals can exist in non-integer dimensions.
For example, we can consider the Sierpinski Triangle, which looks like this:
If we accept our original ideas about dimensions to be valid, then the dimension of the Sierpinski Triangle - which, by our definition, exhibits a fractal pattern - would be governed by:
Fractals have an “unusual relationship with space” because they - as shown by the Sierpinski Triangle - can exist in non-integer dimensions. Amazing, right?
The three bottom images above are taken from the Mandelbrot Set - which has a topological dimension of 2 (meaning that it’s visualised, and pictured, in two dimensions) but has a fractal dimension that’s much more complex. Fractals from the Mandelbrot set - and other fractal patterns - are found throughout nature, including in the growth patterns of bacteria at the top of this post! Other examples include unfurling ferns, and the inside of certain types of seashells. It’s even been shown that the small motions of our eyes follow a fractal pattern - which is why we find things that exist in a fractal pattern aesthetically pleasing.
Next time you see a fractal pattern, remember that it doesn’t just exist topologically as you see it - it’s fractal dimension could be 1.58, or 1.7! The maths behind fractal images are just as beautiful as the images themselves. In fact, in my opinion they enhance the beauty of the images my eye naturally finds aesthetically pleasing.
Images: The top two images shown are of bacteria cultures exhibiting fractal growth due to environmental stress. The bottom three images are from the Mandelbrot Set.
In 1916, Albert Einstein revolutionized the physics world with his theory of general relativity. This theory was the first to predict the existence of gravitational waves - a fascinating concept. Gravitational waves are effectively ripples in the curvature of spacetime which travel outward from the source - sources could possibly include binary star systems composed of white dwarfs, neutron stars or black holes. Gravitational waves cannot exist in the Newtonian theory of gravitation, since in it physical interactions propagate at infinite speed.
Einstein’s theory of general relativity effectively states that gravity is a phenomenon due to the curvature of spacetime. Massive objects cause this curvature - with mass being roughly proportional to the strength of the curvature that object produces. As massive objects move around in spacetime, this curvature inevitably changes. In general, gravitational waves are produced by objects whose motion include acceleration and are not symmetric (examples of symmetrical motion would be an expanding balloon or spinning cylinder). When accelerated, these objects would cause disturbances in spacetime which would spread like ripples on the surface of a pond. This disturbance is known as gravitational radiation - which is thought to travel at the speed of light and never stop or slow down, yet weaken with distance.
Although gravitational radiation has not been directly detected, there is indirect evidence for its existence. The 1993 Nobel Prize in Physics was awarded for measurements of the Hulse-Taylor binary system, which suggests that gravitational waves are much more than mere mathematical anomalies. gravitational wave detectors exist, yet they remain unsuccessful in detecting such phenomena.