At-home STEM Activities: Geometric Mandalas
Geometry is one of the oldest branches of mathematics, and it’s concerned with studying shapes and figures in the world around us. Since geometric concepts include measuring lengths, areas, and volumes, the study of geometry arose independently in many different cultures as they built their civilizations. The ancient Greek mathematician Euclid is often referred to as “the father of geometry,” since he formalized the branch of mathematics by documenting all of his geometric axioms and theorems in his book, Elements.
One prominent part of Elements is Euclid’s constructions. Classic constructions utilize only a compass and a straightedge, and using these two tools, we can create many geometric figures.
Euclid, as shown in raphael’s “School of athens,” demonstrating a construction with a compass to his students
Geometry is applied in many fields outside of mathematics, including physics, architecture, and art. One example of the meeting of art and geometry is a mandala. Mandalas are geometric designs that often rely of symmetry around a circle. In various spiritual traditions, mandalas are used as an aid in meditation. The beauty of mandalas lies in their symmetry, and the creation of mandalas is often associated with restoring inner peace.
Cosmological mandala with Mount Meru, silk tapestry, China via The Metropolitan Museum of Art
Buddhist monks creating a sand mandala (image via mymodernmet.com)
Mayan Tzolk'in wheel from 498 AD (Image via wikimedia)
south rose window in notre dame cathedral in paris (image via wikimedia)
Let’s create our own geometric mandalas using Euclid’s constructions!
Geometric Mandalas
Let’s first learn a few basic constructions. For these, you’ll need a straightedge (like a ruler) and a compass. If you don’t have a compass, you can make your own.
DIY Compass
Materials:
2 pencils
Straight, rigid object (this can be a third pencil, a pen, a dowel, etc.)
Rubber bands
Instructions:
1. Wrap one rubber band around the top of the pencils.
2. Attach the rigid object to each pencil, using two more rubber bands.
3. Done! You can adjust the distance between the pencils along the rigid object to make different sized circles.
Constructions:
Equilateral Triangle
An equilateral triangle is a triangle with sides of equal length
Start with a straight line. Set the compass to the length of the line.
Draw an arc from each end point of the line. Mark the intersection of the arcs.
Connect the intersection point with the two endpoints, forming the triangle.
Square Inscribed in a Circle
Start with a circle and draw a line through the middle.
Set the compass to the length of the line and draw an arc from each end point of the line.
Draw a line through the intersection points of the arc through the middle of the circle. Mark the points of the lines where they intersect the circle.
Connect the points to form a square.
Regular Hexagon
A regular hexagon has sides of equal length
Start with a circle. Choose any point on the circle and draw a circle of the same size centered on that point.
Mark where the second circle intersects the first (either of the two points). Draw another circle of the same size centered at one of these points.
Repeat this process until you have six circles around the original circle. Mark where each of these circles intersect the original circle.
Connect the intersection points to form a hexagon.
As an additional design element, we can use a basic concept from calculus, which is that curves can be made from straight lines.
Start with a square, this one is 4 inches-by-4 inches. Mark off 0.25 inch distances along each side.
Now connect the marks with straight lines. Starting with the left and bottom sides of the square, connect the top mark on the left side to the furthest left mark on the bottom side.
Continue connecting the marks, moving down the left side and to the right on the bottom side.
Repeat this process with the top and right sides of the square to form a leaf shape in the middle of your square (bottom middle photo below)
Repeat this process with all adjacent sides to form the figure in the bottom right photo below.
