## 1 April 2023 (Fiction)

As the cost of fabricating high-fidelity, quantum-collapsed geometry becomes increasingly prevalent, as an industry, we’re forced to confront the following challenge: where are we going to put all of our crap? At QE3D, we’re hard at work to collapse wave functions in our commitment to enable more engineering design space.

You may wonder: how does the superposition of quantum entanglement and machine learning scintillate more room for our everyday carry? Indeed, our technology achieves for consumer products, implantable electronics, wearable devices, and sub-dermal surveillance exactly the same advantages parachute pants achieved for break dancers. With the supremacy of the mesoscale fully realized via TPMS, spinodal decomposition, and mixed topology lattices, from what extra space might we draw additional engineering acumen?

At QE3D, we manifest our quantum technology through three regimes for AI-driven, dimensionality enhanced, spatial domains.

### The geometric fourth dimension

Daniel Piker introduced me to the concept of transforming otherwise Euclidean three-dimensional objects via stereographic projection onto the Riemann hypersphere. Astonished by the myopia of the three.js community, I launched four.js, which has mostly enjoyed success in the dimension orthogonal to this reality.

### Exploiting the Hausdorff dimension of fractal boundaries

Throwing differential geometry out the window, we apply variational analysis to fractal boundaries with fractional Hausdorff dimension, creating a countable class of nooks and crannies. We can apply Stokes equations using Monte Carlo techniques, which speak the language of quantum. This approach is particularly useful when combined with topology optimization, as realized in this lovely tufted furniture collection by EvilRyu

### Conformal maps into hyperbolic embeddings

Recall that in a Riemannian manifold, which in infinitely differentiable, the curvature at any point maybe positive negative, or hyperbolic. As we increase in dimension, the hyperbolic case becomes more prevalent. This trend offers the opportunity to put that extra space to work! On a hyperbolic point of a surface, for example, a circle drawn around that point has a larger circumference that the same radius circle on a flat (Euclidean) surface, which is in turn larger than a circle of the same radius in spherical space. That larger circumference gives us more space to place objects, and the same is even more true as you increase dimension.

For example, consider the circles on this hyperbolic surface from Keenan Crane’s course on discrete differential geometry.

In any number of dimensions, we can use the sum and difference fields of two distance fields \(\df{A}\) and \(\df{B}\):

\[\df{A}+\df{B} \;,\]and:

\[\df{A}-\df{B} \;,\]to create conformal maps between any two sets represented by \(\df{A}\) and \(\df{B}\), such as the conformal map between this square and circle:

### Conclusion? Confusion!

If you’ve been working in boring old 2D or 3D CAD in Euclidean spaces, you’re probably leaving a lot of performance on the table. At QE3D, our software and slicing stack deconvolves bloated designs directly into high-dimensional, fractional-dimensional, and hyperbolically curved finite matter arrays. Join our wait list today!