Abstract

Art, Math and Science at first seem like very different subjects, but they are related in variety of ways. Experience the relationship between math and art by creating beautiful curves using series of straight lines.

Line Art

Make Curves Using Straight Lines

Connect regularly spaced dots using a ruler to create geometric patterns.  Systematically placed straight lines can produce outlines of beautiful curves.

Objectives:

  • Experience the relationship between math and art.
  • Explore the math behind line art.

Materials and Tools:

  • Pencil
  • Eraser
  • Ruler
  • Line art template (at the end of this document)
  • Colored pencils. (Optional).

Steps:

  1. Draw a line from the farthest mark from the origin on the y axis (vertical line) to the closest mark to the origin on the x axis (horizontal line).
  2. Connect the 2nd farthest mark on the y axis to the 2nd closest mark on the x axis.
  3. Continue connecting lines between the points by moving down on the y axis and across the x axis.
  4. The resulting curve is a beautiful parabolic arc.

Variations

  • For younger children, reduce the number of points on the paper.
  • Label points with numbers to show which points to connects together.
  • Glue the template to a piece of cardboard for curve stitching
  • Use coloring pencils.
  • Use a different line art template (see resources) or create your own.
  • Use two other sides to create a second , third or fourth curve.
  • Use a different method to connect dots.

Art, Math and Science at first seem like very different subjects, but they are related in variety of ways. Patterns and structures are often found in artistic representation and scientific study.  Patterns are also fundamental to mathematics. Many scientists and mathematicians study shapes and patterns to understand the world we live in. Artists often study the world we live in and represent it with shapes and patterns they find.

Additional Resources:

Other ways to make line art: http://mathforum.org/library/drmath/view/56710.html

More beautiful envelopes: http://mathforum.org/mathimages/index.php/Envelope

Doodling in Math Class: https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart

More line art templates: http://www.muminthemadhouse.com/maths-and-art-collide-parabolics-curves/

Mathematical Background

A set of lines that share a common equation is called a “family” of lines, and the curve a family of lines create is called an “envelope”.

If we use a regular xy coordinate as a template for this line art activity, then we find the numbers on x and y axis where each line cross always add up to the same value., So we have a “family” of lines, and we can describe all lines using just one equation.

Scaling the sum value to 1, each line has x and y intercepts of (t, 0) and (1-t, 0)  (t is any number between 0 and 1).  So the line equation in intercept form can be written as

The envelope of a one-parameter family of curves given implicitly by F(t,x,y) can be found by solving

==========

So what is the equation of the curve that appears when we connect corresponding marks of our template?

Take the equation that describes our family of lines

And clearing the fractions and rewriting our line equation, it becomes a quadratic equation in “t”,

For every point (x, y) on the straight lines, there is a real number value for “t” . Since both F and F’ are 0 on the boundary points on the curve, t there has to be a double root of F(t, x, y).

So the equation for the envelope curve can be found by setting discriminant of the function to 0.

This is an equation for a tilted parabola.

References:

http://www.ams.org/samplings/math-behind-the-curve.pdf

https://plus.maths.org/content/bridges-string-art-and-bezier-curves

http://mathworld.wolfram.com/Envelope.html

Fleet Science Center

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