New Dimensions

You may have heard its name if you're a Marvel Comics Fan or if you've watched movies from the Marvel Cinematic Universe (MCU) & Sci-fi movies where this name was featured. Apart from that, if you're a student of Physics or Mathematics, it might be as common to you as sugar CUBES (pun intended). Other than that in layman's term it might sound like a horrible virus, disease, an unknown Latin word for something & what not! Trust me, it's not what you think & even not anything you can possibly think of. Though it might sound a bit complicated, I will try my level best to make it as easy and comprehensible as possible. Let the drum roll begin... I'm talking about 'Tesseract'.

A 3D projection of an 8-cell performing a simple rotation about a plane which bisects the figure from front-left to back-right and top to bottom

You would ask, What in the world is that? Actually, it's not from this world, literally! A Tesseract is a representation of the Fourth Dimension just like a cube represents the Third Dimension. In geometry, the tesseract is the four-dimensional analog of the cube; the tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of six square faces, the hypersurface of the tesseract consists of eight cubical cells. The tesseract is one of the six convex regular 4-polytopes. In elementary geometry, a polytope is a geometric object with 'flat' sides.

  
According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek τέσσερεις ακτίνες (téssereis aktines, "four rays"), referring to the four lines from each vertex to other vertices. In this publication, as well as some of Hinton's later work, the word was occasionally spelt 'tessaract'.
A tesseract is bounded by eight hyperplanes. In geometry, a hyperplane is a subspace of one dimension less than its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if a space is 2-dimensional, its hyperplanes are the 1-dimensional lines. Each pair of non-parallel hyperplanes intersects to form 24 square faces in a tesseract. Three cubes and three squares intersect at each edge. There are four cubes, six squares, and four edges meeting at every vertex. All in all, it consists of 8 cubes, 24 squares, 32 edges, and 16 vertices.
If the idea of Tesseract is mind-boggling to anyone from the outset or if any of our geeks like to have an overview about it, I would suggest you all, to watch the two videos for which I'm providing the links down below. I hope it will give you all a basic concept about Tesseract and help in understanding this blog.
Explanation of Tesseract and 4th Dimension by Carl Sagan
Understanding 4D- Tesseract

A diagram showing how to create a tesseract from a point

An animation of the shifting in dimensions as shown above

The construction of a hypercube can be imagined the following way:

• 1-dimensional: Two points A and B can be connected to a line, giving a new line segment AB.
• 2-dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.
• 3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.
• 4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.

It is possible to project tesseracts into three- and two-dimensional spaces, similarly to projecting a cube into two-dimensional space. Projections on the 2D-plane become more instructive by rearranging the positions of the projected vertices. In this fashion, one can obtain pictures that no longer reflect the spatial relationships within the tesseract, but which illustrate the connection structure of the vertices, such as in the following examples:

A tesseract is in principle obtained by combining two cubes. The scheme is similar to the construction of a cube from two squares: juxtapose two copies of the lower-dimensional cube and connect the corresponding vertices. Each edge of a tesseract is of the same length. This view is of interest when using tesseracts as the basis for a network topology to link multiple processors in parallel computing: the distance between two nodes is at most 4 and there are many different paths to allow weight balancing.

The tesseract can be unfolded into eight cubes into 3D space, just as the cube can be unfolded into six squares into 2D space. An unfolding of a polytope is called a 'net'. There are 261 distinct nets of the tesseract. The unfoldings of the tesseract can be counted by mapping the nets to paired trees (a tree together with a perfect matching in its complement).

Since their discovery, four-dimensional hypercubes have been a popular theme in art, architecture, and fiction. Notable examples include:

• Crucifixion (Corpus Hypercubus) – Oil painting by Salvador Dalí featuring a four-dimensional hypercube unfolded into a three-dimensional Latin cross.

The Dali cross, a net of a tesseract

• The Grande Arche – A monument and building near Paris, France said to resemble the projection of a hypercube.

Grande Arche (La Grande Arche de la Défense) Paris, France

• "And He Built a Crooked House" – A science fiction story featuring a building in the form of a four-dimensional hypercube written by Robert Heinlein (1940).
• Fez (video game) - A game where you play as someone who can see beyond the two dimensions other characters can see and must use this ability to solve platforming puzzles. Features "Dot," a tesseract who helps you navigate the world and tells you how to use abilities, fitting the theme of seeing beyond typical dimensions.
• In Madeleine L'Engle's novel A Wrinkle in Time, the characters in the story travel through time and space using tesseracts. The book actually uses the idea of a tesseract to represent a fifth dimension rather than a four-dimensional object (and also uses the word 'tesser' to refer to movement from one three dimensional space/world to another).
• In the science fiction novel Factoring Humanity by Robert J. Sawyer, a tesseract is used by humans on Earth to enter the fourth dimension and contact another civilization on a planet orbiting the star Alpha Centauri A. The hypercube initially exists as a series of connected 3-dimensional cubes, which represent a hypercube that has been unfolded. Refolding the cube in a certain specific manner causes the reformation of the hypercube in 4 dimensions.
• In John Mighton's play, Half Life, one of the characters (an ageing mathematician) builds a tesseract (or rather, the projection of a tesseract) out of popsicle sticks.
• In the Season 1 episode 'Rampage' of the television crime drama NUMB3RS, main character mathematician Charlie Eppes discovers a popsicle-stick tesseract (projection) he built as a boy.
• In Marvel comics and MCU movies, though not the actual tesseract's geometry, The Tesseract (also called the cube) is a crystalline cube-shaped containment vessel for the Space Stone, one of the six Infinity Stones that predate the universe and possesses unlimited energy.

The Tesseract (Cube) which serves as a vessel for The Space Stone, in Marvel comics & MCU

• In the movie Interstellar by Christopher Nolan, there is a reference to the Tesseract, even though it was referred to as a fifth-dimensional one, the protagonist- Cooper enters into one inside a black hole. If you ask me, this is my favourite and probably one of the best representation of the 'Tesseract' & Kudos to Nolan and his team for showcasing it on screen with immaculate details.

Cooper in the Tesseract, from the movie Interstellar

  

Now one would ask, why should anyone study or spend their time or resources in it? This one looks like a valid question as the topic of our discussion is currently imaginary. But one should consider the immense potential in it as it can benefit us in more ways than one. The Tesseract can open the possibilities of teleportation, even on a galactic level, time travel, & even unravel the secrets for the existence of God, as many believe Gods to be nothing but creatures of a higher dimensional space. Hence, if we can get at least closer to The Gods, lest connecting or even interacting with them, only God knows (no pun intended) what we can accomplish in the future. It's almost next to impossible to even imagine or perceive anything in the Fourth Dimension because we are Three Dimensional creatures. So even if there exists 4D space or creatures, chances are we might never come across it. Therefore, it can be a bit overwhelming and difficult to comprehend this concept for someone who has absolutely no idea about it. But I've tried my best to make this topic as understandable as possible. I hope by now the concept is a bit clear & what it is, if not please let me know in the comments if there's any question or any doubts. Finally, I would like to end this blog with two fitting quotes from Madeleine L'Engle's novel, A Wrinkle in Time-"Speaking of ways, pet, by the way, there is such a thing as a tesseract." & "Just because we don't understand doesn't mean that the explanation doesn't exist."

A 3D projection of a tesseract performing a double rotation about two orthogonal planes