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Related Concept Videos

Surface Tension01:24

Surface Tension

Surface tension is defined as the force per unit length (γ) acting along the surface of a liquid. It arises due to strong intermolecular forces of attraction. A molecule located inside the bulk of the liquid is surrounded by other molecules and experiences equal forces in all directions. However, a molecule at the surface experiences unbalanced forces because there are more neighboring molecules below than above. This creates a net inward force that pulls surface molecules toward the interior,...
Surface Tension and Surface Energy01:16

Surface Tension and Surface Energy

When a paint brush is immersed in water, the bristles wave freely inside the water. When it is taken out, the bristles stick together. The reason behind this effect is surface tension.
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Oriented Surfaces

A surface is called orientable if a consistent choice of unit normal vector can be made at every point on the surface. A thin soap film stretched across a wire loop provides a familiar example. The film separates the air on one side from the air on the other, so one side can be selected as positive and the opposite side as negative. Once this choice is made, a unit normal vector can be assigned smoothly across the entire surface.At each point on the soap film, a unit normal vector points...
Quadric Surfaces01:28

Quadric Surfaces

Quadric surfaces are three-dimensional surfaces characterized by second-degree equations in the variables x, y, and z. These surfaces are smooth and continuous, and specific combinations of squared and linear terms define their shapes. The main types of quadric surfaces include ellipsoids, cones, paraboloids, and hyperboloids. Each type exhibits distinct geometric features depending on how the variables are arranged and related within the equation.Ellipsoids are closed surfaces formed when all...
Bending of Curved Members - Neutral Surface01:16

Bending of Curved Members - Neutral Surface

In curved beams, unlike straight beams, the stress distribution across the cross-section is not uniform due to the beam's curvature. This non-uniformity arises because the neutral axis, where stress is zero, does not align with the centroid of the section. In a curved beam, the strain varies along the section as a function of the distance from the neutral axis.
Consider the curved member described in the previous lesson. According to Hooke's law, which relates stress to strain within the...

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Collapsible scissored surfaces.

Noah Toyonaga1, Seri Nishimoto2, Colter J Decker3

  • 1Department of Physics, Harvard University, Cambridge, MA 02138.

Proceedings of the National Academy of Sciences of the United States of America
|June 18, 2026
PubMed
Summary
This summary is machine-generated.

Researchers developed a new method to design transformable pantograph lattices. These structures unfold from 1D to 2D surfaces, enabling novel applications through automated 3D printing.

Keywords:
geometric metamateriallinkagemechanismorigami

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Area of Science:

  • Mechanical Engineering
  • Materials Science
  • Computational Design

Background:

  • Transformable structures offer unique mechanical properties.
  • Existing designs often lack scalability and design flexibility.
  • Pantograph mechanisms provide a basis for deployable structures.

Purpose of the Study:

  • To introduce an additive design approach for pantograph lattices.
  • To demonstrate the unfolding capabilities of these lattices into 2D surfaces.
  • To develop an automated fabrication method for complex pantograph structures.

Main Methods:

  • An additive design approach using two-bar linkages (scissor mechanisms).
  • An algorithm for growing pantograph structures to explore design space.
  • Computational design and physical assembly of example lattices.
  • Streamlined automated fabrication using multimaterial 3D printing.

Main Results:

  • Pantograph lattices can transition from 1D collapsed states to 2D surfaces.
  • The developed algorithm enables exploration of diverse pantograph lattice designs.
  • Successfully designed and assembled complex pantograph structures.
  • Demonstrated efficient automated fabrication via 3D printing.

Conclusions:

  • The additive approach facilitates the design of versatile transformable pantograph lattices.
  • Automated fabrication enables rapid prototyping and realization of complex deployable structures.
  • This work opens avenues for novel applications in adaptive architecture and robotics.