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Theorems of Pappus and Guldinus: Problem Solving01:12

Theorems of Pappus and Guldinus: Problem Solving

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Pappus and Guldinus's theorems are powerful mathematical principles that are used for finding the surface area and volume of composite shapes. For example, consider a cylindrical storage tank with a conical top. Finding the surface area or volume can be challenging for such complex shapes. These theorems are particularly useful in calculating the volume and surface area of such systems. Here, the cylindrical storage tank with a conical top can be broken down into two simple shapes: a...
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Mohr's Circle for Moments of Inertia: Problem Solving01:14

Mohr's Circle for Moments of Inertia: Problem Solving

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Mohr's circle is a graphical method for determining an area's principal moments by plotting the moments and product of inertia on a rectangular coordinate system. This circle can also be used to calculate the orientation of the principal axes.
Consider a rectangular beam. The moments of inertia of the beam about the x and y axis are 2.5(107) mm4 and 7.5(107) mm4, respectively. The product of inertia is 1.5(107) mm4. Determine the principal moments of inertia and the orientation of the major and...
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Area Computation by the Alternative Coordinate Method01:24

Area Computation by the Alternative Coordinate Method

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The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
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Mohr's Circle for Moments of Inertia01:10

Mohr's Circle for Moments of Inertia

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Mohr's circle is a graphical method to determine an area's principal moments of inertia by plotting the moments and product of inertia on a rectangular coordinate system.
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Parallel-Axis Theorem for an Area01:12

Parallel-Axis Theorem for an Area

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The moment of inertia is a fundamental concept in mechanical engineering that plays a significant role in designing rotationally symmetric objects such as flywheels, gears, and other mechanical systems. In this context, we will discuss the moment of inertia of a flywheel rotating about its centroidal axis and how it relates to the moment of inertia about an axis parallel to it.
For a flywheel approximated as a solid disc, consider an infinitesimal differential element with an arbitrary distance...
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Centroid of a Body: Problem Solving01:03

Centroid of a Body: Problem Solving

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The centroid of a body is a crucial concept in engineering and physics. Finding the centroid of a body can help determine its stability, its balance point, and even its design. In this context, consider a thin wire bent in the form of a quarter circular arc. Polar coordinates are used to calculate the centroid. The wire is first divided into small differential elements of a length equal to the radius multiplied by the differential angle.
The x-coordinates and y-coordinates of each element's...
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Related Experiment Video

Updated: Oct 14, 2025

An Efficient and Flexible Cell Aggregation Method for 3D Spheroid Production
07:46

An Efficient and Flexible Cell Aggregation Method for 3D Spheroid Production

Published on: March 27, 2017

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Improved interval methods for solving circle packing problems in the unit square.

Mihály Csaba Markót1

  • 1Faculty of Mathematics, Wolfgang Pauli Institute, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Journal of Global Optimization : an International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering
|November 1, 2021
PubMed
Summary
This summary is machine-generated.

Computer-assisted proofs confirm the densest packings for 31-33 equal circles in a square. Enhanced methods significantly accelerated solutions, achieving results in hours instead of months.

Keywords:
Branch and boundCircle packingGlobal optimizationInterval arithmeticOptimality proof

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Area of Science:

  • Computational Geometry
  • Optimization Algorithms

Background:

  • Previous work established computer-assisted methods for circle packing problems up to 30 circles.
  • Solving larger instances (31-33 circles) required significantly more computational resources with existing techniques.

Purpose of the Study:

  • To develop and apply improved computational methods for proving optimal circle packings.
  • To solve the densest packing problems for 31, 32, and 33 equal circles in a square.

Main Methods:

  • Developed a new interval-based polygon representation for the local search phase, enhancing efficiency and correctness.
  • Implemented improved strategies for the global search, including symmetry filtering and tile pattern matching.
  • Utilized interval arithmetic and global optimization techniques for rigorous proof generation.

Main Results:

  • Successfully determined the densest packings for 31, 32, and 33 non-overlapping equal circles in a square.
  • Achieved optimal solutions in significantly reduced computation times (26, 61, and 13 CPU hours).
  • The new methods are approximately 40-100 times faster than previous approaches.

Conclusions:

  • The enhanced computational approach provides highly precise enclosures for optimal circle packing solutions.
  • The improved methodology is scalable and capable of solving subsequent, larger circle packing instances efficiently.