In this page, the students were introduced to the basic principles of geometric contructions and were given a step-by-step manual which would help them understand the way of work they had to adopt in order to identify and reproduce geometry patterns. This document was co-written by the Mathematicians involved in the project and the techniques explained here were used by the students in a number of tasks throughout the project. Therefore, this page was a point of reference for all the students involved.
Using our imagination and the principles of geometric constructions in an underlying geometric framework we will be able to create the patterns that we come across in our surroundings.
When we say geometric construction according to Euclidean Geometry we mean construction by using only a straightedge and a set of compasses. By using them, we can create many geometric shapes and patterns.
Most of the patterns which we meet in our surroundings are a combination of geometric shapes (circles, polygons, lines), which are repeated. So we can create underlying grids, which are structure serving as the foundation upon which we can construct patterns.
Let us do the following:
Activity 1: Seven overlapping circles
Materials: Paper, Straightedge, Compass, Pencil-Pen.
- Using a straightedge, draw a horizontal line near the center of the paper.
- Make a circle with the compass point placed near the center of the line. Using the intersection points as new compass points, draw a circle on either side of the first circle.
- Add four more circles using the new points of intersection as compass points. It is important that all circles have the same radius.
Activity 2: Creating Triangle and Hexagon Grids
Materials: Straightedge, two different colored pens, tracing paper.
Creating Triangle and Hexagon Grids
In activity 1, if you had continued adding overlapping circles at the intersection points, the result would be a circle grid as shown in the seven overlapping grid.
figure 2.1
1.On the seven-overlapping-circles grid (figure 2.1), place a dot at the center of each rosette (figure 2.2).
figure 2.2
2. Place the tracing paper over the circle grid, and connect the dots in horizontal and diagonal lines to make a triangle grid (figure 2.3).
figure 2.3
3. Using a different color of pen, mark the hexagon grid by highlighting the outer edge of six adjoining triangles as shown (figure 2.4).
figure 2.4
Activity 3: Five Overlapping Circles
Materials: Paper, Straightedge, Compass, Pencil-Pen.
1. Use a straightedge to draw one horizontal and one perpendicular line. Mark the point of intersection as A. Place the compass point at point A and draw a circle. Mark the points that cross the lines B, Γ, Δ and E
2. Using points Β, Γ , Δ and E draw four more circles. All circles must have the same radius.
Activity 4: Creating Square Grids from Circles
Materials: Straightedge, three different colored pens, two pages with five overlapping grid of activity 4, tracing paper.
Creating a five- overlapping- circles grid
In activity 3, if you had continued adding overlapping circles at the intersection points, the result would be a circle grid as shown in figure 4.1
figure 4.1
- Place a dot at the central point of each rosette.
figure 4.2
2. With the tracing paper over the circle grid, connect the dots horizontally and vertically to make a square grid (figure 4.3).
figure 4.3
3. Using the straightedge and a different colored pen, draw diagonal lines (figure 4.4)
figure 4.4
4. Erase the horizontal and perpendicular lines. With the tracing paper over the circle grid, connect the points of intersection of the diagonal lines horizontally and vertically using the straightedge and a different colored pen (figure 4.5)
figure 4.5
Activity 5: Seven overlapping circles
Materials: Paper, Straightedge, Compasses, Pencil-Pen.
- Use a straightedge to draw one horizontal and one perpendicular line. Mark the point of intersection as A. Place the compass point at point A and draw a circle.
2. Using the center of the first circle, draw a second circle with a smaller radius.
3. Draw a third circle smaller than the previous with its center point on the perpendicular line and tangent on the second circle.
4. Using the center of the third circle make a circle with a smaller radius than the third circle and tangent on the first circle.
5. Draw one more circle with a radius equal to the radius of the first circle, its center near the perpendicular line and tangent on the fourth circle.
6. Using the center of the fifth circle make a circle with a radius equal to the one of the second circle and tangent on the fourth circle.
7. Repeat the steps 3, 4, 5 and 6 on the horizontal line.
8. Repeat step 7 so as to form a square with the circles as shown in the figure below.
9. If you had continued adding circles, the result would be a circle grid as shown in the figure below.
