fairfolio
Saved Projects

An Investigation of Shapes of Unvarying Height

CSEF · 2013 Mathematics & Software

Zachary H. Seligman Karen · Los Angeles County High School for the Arts, Los Angeles, Los Angeles County

Overview

Objectives/Goals This research seeks to examine the characteristics and traits of shapes of unvarying heights regardless the orientation. Through simple proofs using geometric/trigonometric principles, I derived the general formulas for the areas and perimeters of Reuleaux Polygons, constructed from an n-gon with an odd number of sides. Furthermore, to observe their general behavior as the number internal sides approaches infinity, the equations derived were evaluated with limits with cases of L#Hopital#s Rule. Methods/Materials The construction is simple; using the longest diagonal from each of the vertices on any regular, odd-sided n-gon as a radius to create an arc. Results The perimeter of these shapes is based on the length of the arcs, the height and number of internal sides being the only variables. The area is particularly interesting as it encompasses both the area of the regular, odd-sided n-gon and the area of the little "smigums" allowing for the regular height. "Smigums" being defined as the difference between a sector and its corresponding internal triangle. Also, if the constructed shape is rotated around an axis of symmetry, it creates a three dimensional prism with the property of uniform height too. Conclusions/Discussion Although a circle is the first shape that comes to mind to many people when asked to think of a shape that has a uniform height regardless the orientation, this study showed that infinitely many shapes with a uniform height exist, all based on regular polygons with an odd number of sides.

Summary statement

This project sought to bring light to the behavior and characteristics of Reuleaux Polygons, bringing them into the third dimension, and potential application.

Help received

Father gave guidance in the construction of the standing display board and the 3-dimensional models; Mother made sure I was fed so I could think properly; Annabel Adriatico gave tips on general logistics for the writing of the project; Nicolas Abruzzo helped flush out ideas for the direction to take in the

Competition history

  • CSEF 2013 Mathematics & Software · Entry S1421

Resources

Related projects

Closest projects by meaning, across every fair and year in the corpus.

Browse more like this

Source: California Science & Engineering Fair public projects

Save projects to your library

Sign in with Google to keep track of projects you find interesting, organized into folders. An account also raises your daily allowance for “Has this been done?”, and lets you create a key for the MCP server with a much higher limit than anonymous use. Browsing stays public.

Continue with Google