Ironman🤖 a.k.a Tony Stark : "Genius🧠, Billionaire💰, Playboy🐰, Philanthropist🦸🏻♂️". We all know who he is..!
In Ironman 2 - he creates his mini arc-reactor⚡ with some help from his father's journal. But, when we look closely we could find an amazing detail which marvel has left us in one of the pages.. Let's take a look –
At first sight, it seems to be a 4D hyper-cube a.k.a Tesseract, but the most important part lies in the left page – there is an equation🐎! Here is where things get interesting – Obviously, the equation🔢🔣 mentioned there is bit more complex for understanding – let's break that down.
In the set of illustrations, we see shadows of 0D (dot), 1D (line), 2D(square), 3D(Cube) and 4D (hyper-Cube)– so, it is clear that the equation must be something which connects to n-Dimensional cube –
- 0 – A point is a hypercube of dimension zero.
- 1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.
- 2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square.
- 3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.
- 4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract)
We all at some point in time would have come up with the below representation of the hypercube –
Hurray!! our intuition seems to be correct – the equation corresponds to a generalized version for m-dimensional cubes contained in the n-cube–
where,
Other way of simple representation of this equation is by eliminating the boundary of n cubes🔗, here the equation becomes as simple as (x+2)n where n is the dimension – how could a simple equation as this could possibly represent an n-dimensional hypercube? Let's find out!
Let's substitute different values for n and find out what the equation holds in it –
Dot n=0 → (x+2)0 = 1
Line n=1 → (x+2)1 = x + 2
Square n=2 → (x+2)2 = x2 +4x + 4
Cube n=3 → (x+2)3 = x3 + 6x2 +12x+8
Hypercube n=4 → (x+2)4 = x4 + 8x3 + 24x2+ 32x + 16
Here, the powers of x → x4 , x3, x2 and x represents the dimension and if this statement is true, then what does the numbers in the coefficients represent?
Let's again try to correlate with our intuition, for a dot, the number seems to simply represent the number of dots i.e., 1, for a line coefficient of x is 1 hence 1 line (i.e., vertex) and 2 dots (i.e., edge), moving this way for a cube it seems to have 1 cube, 6 squares (faces) , 12 vertices and 8 edges – by graph theory this seems to be all true – Let's us put this into a table and see how it looks –
0 | 1 | 2 | 3 | 4 | |||
|---|---|---|---|---|---|---|---|
n | n -cube | Names | Edge 0-face | Vertex 1-face | Face 2-face | Cell 3-face | 4-face |
0 | 0-cube | Point Monon | 1 | ||||
1 | 1-cube | Line segment Dion | 2 | 1 | |||
2 | 2-cube | Square Tetragon | 4 | 4 | 1 | ||
3 | 3-cube | Cube Hexahedron | 8 | 12 | 6 | 1 | |
4 | 4-cube | Tesseract Octachoron | 16 | 32 | 24 | 8 | 1 |
Once again, the intuition seems to be right!
It's a wonder how a simple equation such as (x+2)n could possibly hold such great information about hyper-cubes! This holds good on higher dimensions. Also, kudos to marvel for adding this level of detail in their movies! Hope you enjoyed the blog as much as I did writing it down!
Have you ever come across equations like these in any other movies and want me to write about it? Let me know!