Close
Close window
  • PRODUCTS

    Maple

    Maple Professional

    Maple Academic

    Maple Student Edition

    Maple Personal Edition

    Maple Player

    Maple Player for iPad

    MapleSim

    MapleSim Professional

    MapleSim Academic

    Online Education

    Maple T.A. - Testing & Assessment

    Maple T.A. MAA Placement Test Suite

    Möbius - Online Courseware

    MapleNet

    Toolboxes & Connectors

    E-Books & Study Guides

    Professional Services

  • SOLUTIONS

    Engineering: Industry Solutions

    Machine Design / Industrial Automation

    Aerospace

    Vehicle Engineering

    Robotics

    Power Industries

    Engineering: Application Areas

    System Simulation and Analysis

    Model development for HIL

    Plant Modeling for Control Design

    Robotics/Motion Control/Mechatronics

    Other Application Areas

    Education

    Mathematics Education

    Engineering Education

    High Schools & Two-Year Colleges

    Testing & Assessment

    Students

    Applied Research

    Financial Modeling

    Operations Research

    High Performance Computing

    Physics

  • PURCHASE

    Purchase & Pricing Details

    Maplesoft Web Store

    Request a Price Quote

    Contact Maplesoft Sales

    Elite Maintenance Program

  • SUPPORT

    Tech Support & Customer Service

    Frequently Asked Questions

    Online Product Help

    Training

    Product Documentation

    Download Product Updates

  • RESOURCES

    Webinars & Events

    Live Webinars

    Recorded Webinars

    Upcoming Events

    Community

    MaplePrimes

    Maplesoft Blog

    Maplesoft Membership

    Maple Ambassador Program

    MapleCloud

    Publications

    Technical Whitepapers

    E-Mail Newsletters

    Maple Books

    Math Matters

    Examples & Applications

    Application Center

    MapleSim Model Gallery

    User Case Studies

    Exploring Engineering Fundamentals

    Teaching Concepts with Maple

    Resources

    Maplesoft Welcome Center

    Teacher Resource Center

    Student Help Center

  • COMPANY

    About Maplesoft

    Contact

    Careers

    Media Center

    Partners

    25th Anniversary Timeline

Application Center

  • Home
  • Editor's Choice Applications
  • MapleSim Model Gallery
  • New Applications
  • Tips & Techniques
  • Contribute your Work

Application Search


  • Advanced Search

Browse Categories

  • Mathematics
    • Precalculus
    • Abstract Algebra
    • Calculus I
    • Calculus II
    • Calculus III
    • Calculus of Variations
    • Chaos Theory
    • Combinatorics
    • Complex Analysis
    • Cryptography
    • Differential Equations
    • Differential Geometry
    • Engineering Mathematics
    • Game Theory
    • General
    • Geometry
    • Graph Theory
    • Group Theory
    • Linear Algebra
    • Logic
    • Number Theory
    • Numerical Analysis
    • Operations Research
    • Optimization
    • PDEs
    • Precalculus
    • Real Analysis
    • Tensors
    • Topology
    • Trigonometry
    • Vector Calculus
  • Computer Science
    • Algorithms
    • C
    • Cryptography
    • Error Correction
    • FORTRAN
    • Graph Theory
    • Internet Connectivity
    • Java
    • Logic
    • Machine Learning
    • Maple Programming
    • MATLAB®
    • Numerical Analysis
    • Theory of Computation
  • Education
    • Algebra
    • Calculus I
    • Calculus II
    • Calculus III
    • Case Studies
    • Complex Analysis
    • Differential Equations
    • Economics
    • Engineering
    • Linear Algebra
    • Maple TA
    • Numerical Analysis
    • Operations Research
    • Quantum Mechanics
    • Real Analysis
    • Statistics
    • Trigonometry
    • Vector Calculus
  • Engineering
  • Finance
  • Maple Tools
    • Internet Connectivity
    • Maple Functionality
    • Maple Programming
    • MapleNet Functionality
  • Physics
  • Science
  • Statistics & Data Analysis

Subscribe

  • New Applications
  • Editor's Choice
Home : User Community : Application Center : Mathematics : Calculus II : Bee-Cell Structure.mw

Bee-Cell Structure.mw

Nature, in general, affords many examples of economy of space and time for anyone who is curious enough to stop for a while and think. Pythagoras (6th Century BC) was reported to have known the fact that the circle is the figure that has the greatest surface area among all plane figures having the same perimeter which is obviously an example of economy of space. Heron (2d Century BC) deduced the fact that light after reflection follows the shortest path, hence an example of economy of space. Fermat (1601-1665) was seeking a way to put the law of light refraction under a form similar to that given by Heron for the light reflection but this time he was looking for an economy of time rather than space. Huygens (1629-1695), building on Fermat finding, considered the path of light not as a straight line but rather as a curve when passing through mediums where its velocity is variable from one point to an other. Hence economy of time. Jean Bernoulli (1667-1748) he too was building on Huygens concept when he solved the problem of the brachystochron which is based on economy of time. This is not to say that nature economy is concerned with only physical phenomena but examples taken from living creatures abound around us. To take one example that many researches have examined very carefully and in many details in the past is that of the honeycomb building in a bee hive. It turns out, as we shall soon prove, that the bottom of any bee-cell has the form of a trihedron with 3 equal rhombi (rhombotrihedron) which, once added to the hexagonal right prism, will make the total surface area smaller resulting in economy on the precious wax that is secreted and used by worker bee in the construction of the entire cell. Our plan in this article has a double purpose: 1- to prove the minimal surface we referred to above using a classical proof. 2- To start with no preconceived idea about the bee-cell then A- to consider a trihedron having 90 degrees dihedral angle between all 3 planes. B- To get an equation relating dihedral angle with the larger angle in a rhombus that, once solved, gives exactly the dihedral angle of 120 degrees along with the larger angle in each rhombus as 109.47122 degrees which are the exact data one can find at the bottom end of a honey-cell. This configuration which is that of a minimal surface of the cell is the only one that our equation can give for all 3 planes to have in common these two angles . All others have different dihedral and larger angles. I believe that the neat and simple equation I arrived at is somehow original and so far I have no idea if anyone else has found it before me.
Learn about Maple

Application Details

Author:
Ahmed Baroudy
Application Type:
Maple Document
Publish Date: February 13, 2020
Created In:
Maple 16
Language: English
Categories:
Mathematics: Differential Geometry
Mathematics: Calculus II
Education: Algebra
Education: Calculus II
Computer Science: Maple Programming
Maple Tools: Maple Programming

Toolkit

Download attached file
Download PDF Preview
Contact the Author
Evaluate Maple

Community Rating:

Your Rating:

 
Move the slider to rate

Tell others about this application!

Tweet



ExclamationPlease note that much of the Application Center contains content submitted directly from members of our user community. Although we do our best to monitor for objectionable content, it is possible that we occasionally miss something. If there is something objectionable on this page, please click here to report it.

Products
  • Maple
  • MapleSim
  • Maple T.A.
  • Professional Services and Consulting
  • MapleNet
  • Toolboxes & Connectors
  • E-books & Study Guides
Industry Solutions
  • Vehicle Engineering
  • Motion Control
  • Power Industries
  • Aerospace
Engineering Applications
  • Plant Modeling for Control Design
  • Virtual Prototyping
  • Real-Time Simulation
  • Optimization & Analysis
Education Solutions
  • Mathematics Education
  • Engineering Education
  • High Schools & 2-Year Colleges
  • Testing & Assessment
  • Students
Applied Research
  • Financial Modeling
  • Operations Research
  • High Performance Computing
  • Physics
Connect & Share
Facebook Twitter YouTube LinkedIn

Maplesoft E-Mail Lists
  • The Maple Reporter
  • Model-Driven Innovation Monthly
  • Other E-mail offerings
Maplesoft Membership
  • Login

Products  |  Solutions  |  Purchase  |  Support   |  Resources  |  Community  |  Company  |  Site-Map  | Login
Language:
English  |  Français  |  Deutsch  |  日本語

      © Maplesoft, a division of Waterloo Maple Inc. 2019. | Terms of Use | Privacy | Trademarks