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Related Concept Videos

Toroids01:27

Toroids

A toroid is a closely wound donut-shaped coil constructed using a single conducting wire. In general, it is assumed that a toriod consists of multiple circular loops perpendicular to its axis.
When connected to a supply, the magnetic field generated in the toroid has field lines circular and concentric to its axis. Conventionally, the direction of this magnetic field is expressed using the right-hand rule. If the fingers of the right hand curl in the current direction, the thumb points in the...
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Mohr's circle is a graphical method for identifying the state of stress at a point in a material, making it easier to analyze stress transformations under plane stress conditions. This two-dimensional technique visualizes both normal and shearing stresses on an element.
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Unsymmetric Bending

Unsymmetrical bending occurs when the bending moment applied to a structural member does not align with its principal axis. This misalignment leads to complex stress distributions and deflection patterns that differ from those in symmetrical bending, and are essential for designing structures to withstand different loading conditions. In unsymmetrical bending, the neutral axis—where stress is zero—does not necessarily align with the geometric axes of the cross-section. The orientation of the...
Torsion of Noncircular Members01:16

Torsion of Noncircular Members

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Related Experiment Video

Updated: May 14, 2026

Fabrication of Three-Dimensional Graphene-Based Polyhedrons via Origami-Like Self-Folding
14:52

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Published on: September 23, 2018

The most efficient origami torus.

Richard Evan Schwartz1

  • 1Department of Mathematics, Brown University, Providence, RI 02902.

Proceedings of the National Academy of Sciences of the United States of America
|May 12, 2026
PubMed
Summary
This summary is machine-generated.

Researchers proved that origami tori cannot have 7 vertices but can have 8 vertices. This finding resolves questions about the most efficient vertex count for constructing these unique geometric shapes.

Keywords:
computer-aided proofoptimal constructionorigami toruspaper torus

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Area of Science:

  • Mathematics
  • Computational Geometry
  • Topology

Background:

  • Origami tori are three-dimensional shapes constructed from triangles meeting at vertices.
  • The sum of angles around each vertex in an origami torus must equal 2π.
  • Previous research explored the existence and properties of origami tori with varying vertex counts.

Purpose of the Study:

  • To determine the existence of origami tori with specific numbers of vertices.
  • To settle the question of the most vertex-efficient origami torus construction.

Main Methods:

  • Utilized geometric and topological principles to analyze origami torus structures.
  • Investigated the constraints imposed by angle sums at vertices.
  • Employed proof-based methods to establish existence and non-existence results.

Main Results:

  • Demonstrated that an origami torus cannot be constructed with exactly 7 vertices.
  • Proved the existence of an origami torus with 8 vertices.
  • Established a boundary for the minimum number of vertices required for origami torus construction.

Conclusions:

  • The non-existence of a 7-vertex origami torus and the existence of an 8-vertex one are confirmed.
  • These results provide definitive answers regarding the most efficient origami torus construction in terms of vertex count.