I think I found a geometric shortcut around the scaling problem

Perhaps the Universe just Bangs? To the reader Crease geometry refers to the mathematical and physical study of how flat surfaces (like paper or thin membranes) transform into 3D shapes through folding along specific lines. In technical fields, this concept is split between the mathematical “blueprints” used in origami and the digital “weights” used to sharpen 3D models in computer graphics.

1. Mathematics of Origami (The “Crease Pattern”) In origami, crease geometry is defined by a Crease Pattern (CP)—a 2D map of all the folds required to reach a final 3D form. These patterns follow strict geometric laws to ensure they can actually be folded without tearing or self-intersecting:

Mountain and Valley Folds: Folds are categorized as “mountain” (convex, pointing up) or “valley” (concave, pointing down). Maekawa’s Theorem: At any point where creases meet (a vertex), the number of mountain and valley folds must differ by exactly two (e.g., 3 mountains and 1 valley). Kawasaki’s Theorem: The sum of alternating angles around a vertex must always equal 180° for the model to fold flat. Two-Colorability: The regions between creases can always be colored with just two colors without two regions of the same color touching, similar to a checkerboard.

Origami: mathematics in creasing Origami Tessellations: From Crease Pattern to Folding … How to Fold from a Crease Pattern – Gathering Folds 2. 3D Modeling and Computer Graphics In digital design tools like Blender or Rhino 3D, crease geometry refers to Edge Creasing, which controls how sharp or rounded an edge appears when a surface is subdivided (smoothed).

Crease Weight: A value (usually 0.0 to 1.0) assigned to an edge. A weight of 1.0 keeps the edge perfectly sharp even after smoothing, while lower values allow it to round off slightly.
Hard vs. Soft Creases: Hard creases create crisp, mechanical edges, while "soft" creases (used in SubD modeling) allow for controlled transitions in more organic shapes.
Controlling Topology: Designers use creases to maintain specific shapes without having to add hundreds of extra polygons, which keeps models "light" and easier to edit.

1. Engineering and Material Science Engineers use crease geometry to design deployable structures—objects that can be folded small for transport and expanded later.

Curved-Crease Geometry: Unlike straight folds, folding along a curve forces the connected surfaces to bend, creating high “geometric stiffness” and unique structural strength. Applications: This geometry is critical for developing NASA’s foldable solar arrays, compact medical stents that expand in blood vessels, and even self-folding robots.

Are you interested in the mathematical theorems behind folding, or are you trying to use crease tools in a specific 3D software?

All About Creasing
Aug 2, 2023 — and we can see that the cage is creating the subdivided geometry in the grain. and that's what we want. but many times you're work...
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Crease Patterns as Art - Robert J. Lang Origami
Sep 28, 2015 — If one takes the narrowest possible dictionary definition of the term, a “crease pattern,” or CP, is nothing more than a set of li...
Robert J. Lang Origami
Modeling the Effects of Creases in an Unfolding Membrane
* Creases are regions of permanent deformation found in thin membranes after folding or. * crumpling, which makes it possible to t...
University of Colorado Boulder

So this is why I failed the Test for advanced placement in the Air Force when I joined?

I remember those tests! Whack!

–Ernst