## Miegakure Update November 2020: art direction in 4D

Development on Miegakure is going well.

Modeling the large buildings is very far along and will be done soon.

I polished many things across the entire game.

In July I recorded my SIGGRAPH 2020 Talk about my Technical Paper on n-Dimensional rigid body dynamics, and will post it publicly soon.

I started working with a new artist to *really* nail down the final look of the game and environments across the whole game in a more integrated way. In the process we finally fully switched the engine to Physically Based Rendering (It was an easy switch, actually, contrary to what could be expected!)… and it makes the game look even better.

These were my thoughts as we nailed down the look of the game, and about how 4D space constrains our art direction in Miegakure:

In Miegakure we procedurally generate many 4D meshes and 3D textures. Just like 3D objects have a surface that is 2D, 4D objects have a surface that is 3D! The game also has many regular 3D meshes (with 2D textures) which are embedded in the 4D world, by giving them 4D thickness.

At first it might seem difficult to generate procedural 3D textures which are as detailed as 2D ones made by hand. And we need both at the same time!

In order to have details that aren’t noisy during transition, for a while I was using a combination of a 2D texture and a 3D texture, where the 2D texture contained more detail but was not affected by the slicing, and was just projected onto the sliced object’s surface. This was a hack, which you can see in the trailers: the high frequency detail of the texture just either slides or streches.

However too much 3D texture detail looks bad during the “transition” anyway. (The transition is what I call the time when the slice rotates 90 degrees after you press the 4D rotate button.) If the slice goes through many tiny objects as it rotates, the time each tiny object will be visible will be very short. This would look like many appearing/disappearing objects. The smaller the objects, the quicker they will be appearing/disappearing. In this video of an MRI of a fruit, the tiny seeds look noisier than the larger overall shape as the slice changes (but the colors are all grayscale and the size is still fairly big so it doesn’t look bad). So if a 3D texture has too much small detail, even if it looks good as a static 2D slice, it will look very noisy during the transition. So actually we don’t want to generate too much 3D texture detail, even if we can!

By the way there is a noise issue in 3D too: when the 3D camera moves over quickly changing detail it can create aliasing (a “shimmering” effect). Much of our 3D handmade content (large buildings, trees…) was already made to be less noisy in that sense. Stylized games have an easier time avoiding this problem since they often contain large flat regions of color.

Also, note that we can replace texture detail by geometric detail. This is part of what happened in the games industry with the transition to Physically-Based Rendering. Textures in PBR are not supposed to contain lighting/shadow information, only material information. For example, a rock texture might just be a simple gray color, and if we want actual cracks in the stone we model them as geometry (or normal maps) instead of dark lines in the texture. One of the goals of PBR is to make sure that the props will look good under many lighting conditions: for example a texture where the dark shadows are already stored in the texture (as opposed to computed using the light source) makes it harder to do that. Here is an example comparison/explanation.

So it is in some sense more correct to use geometric detail instead of texture detail anyway. And most of the time it is simpler to procedurally generate geometry, so!

Miegakure can display more 4D geometric detail now compared to when development started. But there is obviously a similar limit for geometric detail where too much looks noisy.

So we can’t have too much texture/geometry detail, but on the other hand I don’t want the game to have very large flat section of colors like so many games have these days. I think it doesn’t work very well with the dioramas seen from far away, where all the visuals are condensed in a small section of the screen. I think it’s fine for the visuals to be simple if they fill a large area like the entire screen, but if they don’t then it does not give enough interesting stuff to look at.

The slicing mechanic also forces upon the game a certain level of realism. For example the tree canopies need to look good when sliced. We could model the canopy with a small number of large flat planes to give a nice painterly/low-noise vibe. This looks good in a regular game, but not when sliced, because the inner structure of the planes is revealed. It just looks like a bunch of simple intersecting planes instead of many tiny leaves creating a canopy. So we need to model leaves more realistically, but we can always make them mostly the same color to reduce noise, as shown here:

So to summarize: we want detail (enough to look good when sliced, and in dioramas, etc…), but not so much that it looks noisy (in the 3D and 4D sense). Compared to texture detail, geometric detail is easier to make and more correct (in the PBR sense and in how it doesn’t require 4D hacks). The final result is a combination of these constraints. It looks much more polished than before. I can’t wait to show it!

## 4D Toys Version 1.5: Anti-Gravity, Export Mesh for 3D Printing, 4D Accelerometer

Here is another 4D Toys Update (iOS) (Steam VR + Mouse & Keyboard).

It adds three buttons:

• Export Mesh: Exports a 3D mesh of what is currently visible on screen.

These meshes can then be 3D printed, etc… (Unfortunately 4D printing is not yet available in this universe :P).

Find the meshes in your Documents folder. (On iOS you can access it by connecting your iDevice to your computer and using Finder).

Note that Shapeways’ algorithm for filling the inside of meshes sadly fails on some of the coolest meshes like the Star Polychora from the last update, since they are filled with lots of polygons on the inside (btw it’s fun to poke your head inside these meshes in VR!).
• Anti-Gravity: Creates an anti-gravity field that makes objects float. Besides the obvious coolness, this allows to see rotations that are unrestricted from the constraints of the collision with the ground.

• 4D Accelerometer (iOS only): Swaps the forward/backward direction of the accelerometer with the 4th dimension.

That is, tilting your device toward/away from you will result in a force along the 4th dimension instead.

A new page of shapes has been added, with one new level for each button. Visiting each of these levels unlocks the corresponding buttons. Also, all special buttons are now gathered into a new menu button. This lays the groundwork for easily adding new features in future updates!

Btw, the SIGGRAPH conference started and my talk on nD Rigid Body Dynamics is now online (I will post it publicly in a few weeks). The Q&A session is on Wed Aug 26 at 9:00AM PDT!

## SIGGRAPH 2020 Technical Paper: N-Dimensional Rigid Body Dynamics

Excited to announce that my technical paper “N-Dimensional Rigid Body Dynamics” was accepted to SIGGRAPH 2020! Very proud to present research developed for 4D Toys & Miegakure at such a prestigious conference.

Here is the link to the paper and the abstract:

I present a formulation for Rigid Body Dynamics that is independent of the dimension of the space. I describe the state and equations of motion of rigid bodies using geometric algebra. Using collision detection algorithms extended to nD I resolve collisions and contact between bodies. My implementation is 4D, but the techniques described here apply to any number of dimensions. I display these four-dimensional rigid bodies by taking a three-dimensional slice through them. I allow the user to manipulate these bodies in real-time.

Btw I believe it is basically unheard of to have work from an indie game presented in the SIGGRAPH technical papers track?

The paper is full of really fun and beautiful math (obviously Geometric Algebra based, see my recent article) that makes me happy. One reviewer called the work “whimsical,” and they’re not wrong, ahah.

Most of this work (including writing the paper) is from ~2012, but I added a section on the (4D) Dzhanibekov effect at the suggestion of the reviewers. Many thanks to them for helping me greatly improve the paper.

## Interactive Article/Video: Let’s remove Quaternions from every 3D Engine (An Interactive Introduction to Rotors from Geometric Algebra)

I have not yet posted on this blog that last year I released an article/video with interactive diagrams on Geometric Algebra, specifically Rotors. (I also recently updated it). Here is the introduction:

To represent 3D rotations graphics programmers use Quaternions. However, Quaternions are taught at face value. We just accept their odd multiplication tables and other arcane definitions and use them as black boxes that rotate vectors in the ways we want. Why does $\mathbf{i}^2=\mathbf{j}^2=\mathbf{k}^2=-1$ and $\mathbf{i} \mathbf{j} = \mathbf{k}$? Why do we take a vector and upgrade it to an “imaginary” vector in order to transform it, like $\mathbf{q} (x\mathbf{i} + y\mathbf{j} + z \mathbf{k}) \mathbf{q}^{*}$? Who cares as long as it rotates vectors the right way, right?

Personally, I have always found it important to actually understand the things I am using. I remember learning about Cross Products and Quaternions and being confused about why they worked this way, but nobody talked about it. Later on I learned about Geometric Algebra and suddenly I could see that the questions I had were legitimate, and everything became so much clearer.

In Geometric Algebra there is a way to represent rotations called a Rotor that generalizes Quaternions (in 3D) and Complex Numbers (in 2D) and even works in any number of dimensions.

3D Rotors are in a sense the true form of quaternions, or in other words Quaternions are an obfuscated version of Rotors. They are equivalent in that they have the same number of components, their API is the same, they are as efficient, they are good for interpolation and avoiding gimbal lock, etc… in fact, they are isomorphic, so it is possible to do some math to turn a rotor into a quaternion, but doing so makes them less general and less intuitive (and loses extra capabilites).

But instead of defining Quaternions out of nowhere and trying to explain how they work retroactively, it is possible to explain Rotors almost entirely from scratch. This obviously takes more time, but I find it is very much worth it because it makes them much easier to understand!

For example, Quaternions are introduced as this mysterious four-dimensional object, but why introduce a fourth dimension of space to visualize a 3D concept? By contrast 3D Rotors do not require the use of a fourth dimension of space in order to be visualized.

Trying to visualize quaternions as operating in 4D just to explain 3D rotations is a bit like trying to understand planetary motion from an earth-centric perspective i.e. overly complex because you are looking at it from the wrong viewpoint.

It would be great if we could start phasing out the use and teaching of Quaternions and replace them with Rotors. The change is simple and the code remains almost the same, but the understanding grows a lot.

As a side note, Geometric Algebra contains more than just Rotors, and is a very useful tool to have in one’s toolbox. This article also serves as an introduction to it.

And here are some quotes about it:

The clearest explanation of 3D geometric algebra within 15 minutes that I’ve seen so far —BrokenSymmetry
I am sold. While I can understand quaternions to an extent, this way of thinking is a much more intuitive and elegant approach. —Jack Rasksilver
This sets a high standard for educational material, and is a shining example of how we can improve education with today’s technologies. —Sebastien Pierre
When I was in college, I asked one of my math professors why the cross product of two vectors results in a perpendicular vector whose magnitude is equal to the area of the parallelogram formed by the two vectors. Like..what? Why? And what about 2D? They blew me off, and that was a big part of why I stopped taking math in college. […] Anyway, I had pretty much given up on ever truly understanding the whole jumble of seemingly unrelated types that are cross products. But then I saw this: And…wow. Just 15 minutes and a lot more than just cross products suddenly make a lot more sense. —Mason Remaley
I’m a pure math dude at heart, even if I don’t get to do it much any more. Two years ago, my wife asked me, “If you had to get a math equation tattooed on your body, what would it be?” I answered, “i^2 = j^2 = k^2 = ijk = -1”. I felt a brief flush of anger when I saw this headline. This is an extraordinarily good article that should be read by pretty much anyone doing graphics programming. —pflats

I wrote most of in 2011/2012, but didn’t release it because I was not satisfied with part of it. But I thought it was time to let go and release it anyway. I actually think it is hurting the advancement of science that people are still mainly using quaternions instead of Geometric Algebra, so holding on to it was not good.

So last fall/summer I cleaned up some of the diagrams and made a 15 minute long video that follows the article exactly. I never made a video this long, and it was quite exhausting. But I thought it would be really cool to make an article that is perfectly synced to a video, so you can either read it or watch it, and the article serves as an exact table of contents for the video.

I think I came across Geometric Algebra from attending SIGGRAPH a long time ago? Specifically this book: Geometric Algebra for Computer Science by Dorst et al. Later on I found this great book: Linear and Geometric Algebra by Macdonald

Geometric Algebra soon came in handy for Miegakure, specifically to define the 4D equivalent to Quaternions, which I posted about on this blog. Later on it became the backbone of 4D Toys.

Learning about Geometric Algebra was also great because it answered so many questions I had when learning linear algebra, the cross product, quaternions, etc… I basically wrote this article for my past self as a college student.

I recently rewrote the introduction to add more detail about the properties of Rotors and how they relate to quaternions. Even though the content went into detail, it should now be clear what Rotors are from only reading the introduction. I can already see from reading recent comments that it was worth it.

I deliberately picked a cheeky click-bait title…

Something else that might be of interest is the history of Geometric Algebra, so I recently added a heavily summarized version to the end of the article. I think looking at the history makes it clearer how the quaternion viewpoint stayed in people’s minds for longer than necessary…