The Matrix

The Terminator

Background image showing the circles in Klein 4 arrangement

Inversions in Movie Tropes

Introductory Group Theory through Hollywood Movies

Jurassic Park

The Inception

The idea for this post struck on thinking about tropes employed in Hollywood movies and their relationships. I observed a curious inversion between 4 hollywood blockbusters: The Matrix, The Terminator, Jurassic Park, and The Inception. This connection becomes evident when thinking about the narrative tropes empolyed in their world building.

I had been exploring some group theory as a part of my research and I was able to recognize that the kind of inversions taking place between the worlds of these movies can be captured using an algebraic structure called the group. Groups are structures that measure symmetry of objects. But don’t let that intimidate you from understanding the gist of this post. It is easy to follow along if you get the inversion idea between the stories. Let us dive right in and explore this curious connection.

Dual Tropes

A first curious observation was that there’s a duality in the worlds of The Matrix and The Terminator.

The Matrix

In the Matrix movie, humans embedded in the simulation revolt to regain control of the reality where machines are ruling the roost.

The Terminator

In the Terminator movie, machines from the simulation rebel against humans to gain control of the reality from humans.

Thus there is a duality between the worlds of Matrix and Terminator. One has the same source (virtuality) and target (reality), but the outside virtuality in Matrix is inside for Terminator, and the inside of Matrix, reality is outside in the Terminator world.

Arrow Inversions

After posting the tweet, I came across a nice observation from Haig on how Jurassic Park is an inversion of the Matrix.

The Matrix

In the Matrix, humans are trying to collapse the hyperreal simulation they are in, back to reality.

Jurassic Park

In Jurassic Park, we have expanded out the reality to a place where the hyperreal simulation of dinosaurs surround us.

There is an arrow inversion occurring between the two worlds. In Matrix, the motion is from virtuality back into reality, where as in Jurassic Park, it is the inverse movement from reality out to a virtuality of hyperreal dinosaurs. Also, worth noticing is that there is no change in the inside/outside relation of worlds like in the previous case between The Matrix and The Terminator.

An Analogy

This compelled me to complete the following analogy

The Matrix

is to

The Terminator

is as

Jurassic Park

is to

?

Symbolic Algebra

We can leverage our understanding of the accompanying symbols to work this out!

So here, we seek a movie where there is an arrow from the reality outside to virtuality. And this immediately suggests some cool movies with this pattern and a fitting movie is The Inception!

A new dual trope

This lets us see how the worlds in Jurassic Park and The Inception are dual with both the arrows and the worlds reversing their roles, just like in Matrix vs. Terminator situation.

Jurassic Park

In the Jurassic Park, humans in the reality reaches out to the simulated virtuality with the dinosaurs.

The Inception

In the movie Inception, some humans reach in to the virtual world of dreams from the reality.

So this gives us two set of dual worlds in the form of Matrix ⇆ Terminator and Jurassic Park ⇆ Inception.

Converse Worlds

This opens up two new kinds of connections between the worlds which can be thought of as swapping the inside and outside.

Image showing a set of back and forth arrows going between two pairs of movies.

If we keep the arrows constant and instead swap reality with virtuality, we get these converse worlds, where the movement stays constant, but the worlds have been flipped.

Klein 4 group of movies

We can integrate the knowledge we gained by completing the analogy to put them together as something known as a Klein 4 group. This concisely represents the various symmetries present across these movies and the operations of moving across them.

Shows the image of four movies in four corners of a Klein-4 group with horizontal, vertical, and crossing arrows between them. The four corners, in clockwise order, are the movies The Matrix, The Terminator, The Inception, and Jurassic Park.

This gives us 2 pairs of duals, 2 pairs of inverse worlds, and 2 pairs of converse worlds. So all the 6 possible connections between the 4 entities has been captured compactly in this group structure. Group theory is a world of fascinating mathematics where Klein 4 group is but one among a spellbinding array of algebraic structures. Hope this post gave a little taste of what it is like to think about relationships between different entities as a unified structure.