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Tych Brahe Planetarium in Copenhagen

Unit 2 - Conic Sections

The Intersection of Algebra and Geometry

Many buildings incorporate conic sections into their design. Architects have many reasons for using these curves, ranging from structural stability to simple aesthetics. But how can a huge parabola, ellipse, or hyperbola be accurately constructed in concrete and steel? In this Unit, you will learn how the geometric properties of the conics – mixed with algebraic manipulations – are used to construct both monuments and practical structures. In ancient times architecture was considered a integral part of mathematics, so architects had to be mathematicians. Many of the structures they built—pyramids, temples, amphitheaters, and irrigation projects—still stand.

In the modern age, architects employ even more sophisticated mathematical principles. Architects have different reasons for using conics in their designs. For example, the Spanish architect Antoni Gaudí used parabolas in the attic of La Pedrera. He reasoned that since a rope suspended between two points with an equally distributed load (as in a suspension bridge) has the shape of a parabola, an inverted parabola would provide the best support for a flat roof. (View a 360 degree virtual tour of La Padera.​)

Pictured above is the Tycho Brahe Planetarium​ in Copenhagen, Denmark. What geometric shapes can you recognize?

 

Driving Question: How can we use conic sections to design art and architecture?​

Click on a link to jump ahead to the following sections:

  • Lessons
  • Resources


Lessons

  • Lesson 1: Circles and Degenerate Conics
    • 2.1 Investigation - Crop Circles
    • 2.1 GeoGebra App - Full size Interactive conic section
    • 2.1 Practice - Circles - Practice

  • Lesson 2: The Elliptical Universe
    • 2.2 Investigation - Create an animated model of the Solar System
    • 2.1 Video - Elliptical Orbits
    • 2.2 Practice - Ellipses - Practice

  • Lesson 3: Parabola Equations and Graphs
    • 2.3 Investigation - The Human Cannonball
    • 2.3 Practice - Parabola Properties - Practice

  • Lesson 4: Hyperbolic Navigation
    • 2.4 Investigation - The Matrix Online
    • 2.4 Practice - Hyperbola Properties - Practice

  • Lesson 5: Polar Graphs
    • 2.5 Investigation - Nutritional Analysis
    • 2.5 Practice - Linear Systems - Practice

  • Lesson 6: More Polar Curves
    • 2.6 Investigation - Roses, Rings and Hearts...oh my!
    • 2.6 Practice - Polar Curves - Practice

  • Assessment: Desmos Art Project
    • 2.9 UPA Assessment - Desmos Art Project







Resources - Conic Sections

The following are links to resources for Conic Sections:

  • Entire Unit - Coming Soon

  • Matrices - Entire unit on Khan Academy



The following are other free resources:

  • GeoGebra - Free online math tools

  • Desmos - Free online graphing calculator







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