Depth, In Depth: 4D Hologram

Let me guess: you like holograms, you saw the 4D Hologram blog post, and you’re too afraid to ask what any of it means?

I, Nolan, am here today to explain some of the fourth dimensional logic that went into making the fantastical-yet-formulaic mathemagic you can see for free in a Looking Glass.

Welcome to holo-school! Class is in session.

PART ONE: FOURTH DIMENSION IN THEORY

What’s the fourth spatial dimension?

It’s a theoretical extrapolation of our abstract understanding of space according to Euclidean geometry.

What’s a theoretical extrapolation of our abstract understanding of space according to Euclidean geometry?

We typically think about the world in terms of three axes: x (left/right), y (up/down), and z (forwards/back). It’s often useful to compartmentalize space in this way, from giving directions (turn left, go up the stairs) to performing physics calculations (the vertical velocity of a projectile motion depends on gravity). Greek mathematician Euclid famously defined the logical rules that govern this framework, so we refer to it as Euclidean geometry.

It’s an abstraction of reality, but a common and helpful one.

The relationship between these three axes is described as orthogonal (there is a 90-degree right angle between each pair), which means you can represent a point in a three dimensional Euclidean space perfectly using three values, each one a certain distance in a different spatial dimension.

Using this, we can theoretically add another axis (let’s call it “w”), one that is orthogonal to the other three, and define a 4th dimensional point as something that requires four values to perfectly represent it. What does this look like in the real world? We don’t know! We can’t actually build a fourth dimension object, but we can imagine it!

Consider a square. It’s a two dimensional space (area) bounded by one dimensional spaces (lines, or edges). Similarly, a cube is a three dimensional space (volume) bounded by two dimensional spaces (squares, or faces). Continuing, the hypothetical “hypercube” must be a four dimensional space (hypervolume?) bounded by three dimensional spaces (cubes, or hypersurface cells). Woah.

From this overview, you may be able to identify the issues with illustrating a 4D object on flat paper. Even with explanation, it’s very difficult to discern the relative depths of different edges!

PART TWO: FOURTH DIMENSION IN PRACTICE

Why is that hypercube just a bunch of lines?

Viewing a 4D object on a 2D screen is a bit like trying to identify a 3D object by viewing it through a slit in a door. (When you remove two dimensions from a 3D object you get a one dimensional scene, or a line). You lose SO much information it’s almost impossible to make out what you’re looking at. That’s why we use the particular qualities of a holographic display to more accurately render a 4D object and open the door (pun intended) to a whole new dimension of content.

How does 4D Hologram work?

It takes advantage of the Holoplay Unity SDK’s intermediary quilt step. When the Holoplay camera creates a volumetric view of a 3D scene, it makes 45 unique renders (that is, 2D snapshots of the same 3D scene) along a single axis, stores the images in a grid-like “quilt”, then interlaces them together.

If an eye-patch wearing pirate were to stare at the Looking Glass, they would only see one* of those rendered views. It’s only through the use of two viewpoints (eyes) together that the Looking Glass’ primary depth effect, stereoscopy, works. It combines many 2D views together to make a 3D scene.

Instead of taking 2D snapshots of the same 3D scene, 4D Hologram changes the 3D scene for each render based on the camera’s position in relation to the hypercube. Each view is a different 2D representation of the 4D object.

Taken together, this allows the Looking Glass to approximate a transformation in the fourth dimension. It’s still a 3D dimensional view, but it’s a 3D dimensional “slice” of a 4D scene. That is, instead of transforming the camera along the x-axis (as the traditional capture does), 4D Hologram transforms the camera along the w-axis**.

Seeing a hypercube in the Looking Glass gives the viewer a more intuitive understanding of the structure between spatial dimensions. They can use depth perception to sense the distance between three dimensional objects along the w axis, a relationship not easily expressed in two dimensional drawings.

Armed with new knowledge about the fourth dimension, give the 4D Hologram another look! Try to follow a hypersurface cell as it rotates away from you in 3D dimensional space, and contemplate what that means in terms of its position in the abstract fourth spatial dimension. If you still don’t understand what’s going on, that’s ok! It’s all just a bunch of nonsense made up by bored 19th century nerds.

Until next time, deep thinkers!

~Nolan

* Because the angle between views is so small, each eye may actually see parts of multiple views at the same time

** This is a slight oversimplification - the transformation involves a rotation about the XW plane for aesthetic purposes

Nolan Filter is a freelance developer, game designer, and Computer Science professor working on weird cool future tech, like AR, VR, Immersive Displays, and Holograms.