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Related Concept Videos

Euler's Formula for Pin-Ended Columns01:21

Euler's Formula for Pin-Ended Columns

In structural engineering, the stability of columns under compressive axial loads is a critical consideration, described as buckling. A typical example involves a column PQ, which is pin-connected at both ends and subjected to a centric axial load F applied at one end, with a reaction force of F' = -F at the other end. Here, it is crucial to understand that when an applied load exceeds the critical load, buckling occurs as the system becomes unstable.
To calculate the critical load, envision...
The DNA Helix01:16

The DNA Helix

Overview
The DNA Helix01:07

The DNA Helix

Deoxyribonucleic acid, or DNA, is the genetic material responsible for passing traits from generation to generation in all organisms and most viruses. DNA is composed of two strands of nucleotides that wind around each other to form a spring-like structure called a double helix. However, the double helix is not perfectly symmetrical. Instead, there are regularly occurring grooves in the structure. The major groove occurs where the sugar-phosphate backbones are relatively far apart. This space...
The DNA Helix01:16

The DNA Helix

Overview
Newman Projections02:06

Newman Projections

Different notations are used to represent the three-dimensional structure of molecules on two-dimensional surfaces. One of the most commonly used representations is the dash-wedge formula. The dashed wedges, solid wedges, and the plane lines indicate the groups situated behind the plane, coming out of the plane, and in the plane, respectively.
The organic molecules rotate across the single bonds leading to numerous temporary three-dimensional structures of varying energy known as conformers.
Euler's Formula to Columns with Other End Conditions01:15

Euler's Formula to Columns with Other End Conditions

Euler's formula is very important in the field of structural engineering, providing a foundation for understanding the critical loading conditions of pin-ended columns. This formula links the modulus of elasticity, the moment of inertia of the cross-section, and the column's length, offering a precise calculation of the critical load at which a column is prone to buckling.

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Related Experiment Video

Updated: May 28, 2026

Self-assembly of Complex Two-dimensional Shapes from Single-stranded DNA Tiles
10:23

Self-assembly of Complex Two-dimensional Shapes from Single-stranded DNA Tiles

Published on: May 8, 2015

A new Euler's formula for DNA polyhedra.

Guang Hu1, Wen-Yuan Qiu, Arnout Ceulemans

  • 1Department of Chemistry, State Key Laboratory of Applied Organic Chemistry, Lanzhou University, People's Republic of China.

Plos One
|October 25, 2011
PubMed
Summary
This summary is machine-generated.

We introduce a new formula, s + μ = c + 2, for DNA polyhedra (DNA cages). This formula links topological features like components (μ), crossings (c), and Seifert circles (s) to the Euler characteristic, aiding DNA cage design.

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Last Updated: May 28, 2026

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Area of Science:

  • Biochemistry
  • Topology
  • Nanotechnology

Background:

  • DNA polyhedra are complex, cage-like structures formed by interlocked DNA strands.
  • Understanding their topology is crucial for designing novel DNA nanostructures.

Purpose of the Study:

  • To propose a unifying formula for describing the topological features of DNA polyhedra.
  • To establish a theoretical framework for the stereochemistry and design of DNA cages.

Main Methods:

  • Transforming DNA polyhedral links into Seifert surfaces to eliminate knots.
  • Applying Euler's formula to relate topological invariants: components (μ), crossings (c), and Seifert circles (s).

Main Results:

  • A simple formula, s + μ = c + 2, is derived, connecting topological properties.
  • This formula relates the topology of DNA cages to the Euler characteristic of their underlying polyhedra.
  • Seifert circles are identified as effective topological indices for polyhedral links.

Conclusions:

  • The new Euler's formula provides a robust theoretical framework for DNA polyhedra stereochemistry.
  • This framework can aid in characterizing enzymatic DNA transformations and designing novel DNA cages with higher genus.