Other Interesting Mathematical Curves
Spirals

The spiral is a beautiful mathematical curve that we often see around is. The picture below shows a piece of ornamental metalwork, a bracket for a scupture.

The spiral below is a beautiful piece of geometric artwork:

The chambered nautilus seashell is in the shape of a spiral. The creature who lives in the shell has built the chambers of the shell as it grew. The spiral is a curve that often is related to growth outward from the center.

You can construct a mathematical spiral, by following these steps:

Step 1: Construct a square (ABCD) and the midpoint of one side (point E).

Step 2: Using that midpoint (E) as a center, construct a circle passing through the opposite corner of the square (point A).

Step 3: extend side BC until it intersects the circle (at point F). Construct a rectangle (ABFG). this rectangle is called the Golden Rectangle, and the ancient Greeks believed that it's shape was the most pleasing shaped rectangle to the eye, and they called the ratio between the length and width of this rectangle The Golden Ratio.

Step 4: Construct a circle with center G and passing through point D. Where this new circle intersects segment GF is point H. Construct a square DGHI.

Step 5: Construct a circle with center F and passing through point H. Where this new circle intersects segment FB is point J. Construct a square HFJK. Notice that this is the same process as in step 4.

Step 6: Construct a circle with center J and passing through point L. Where this new circle intersects segment JK is point N. Construct a square JLMN. Notice that this is, again, the same process as in step 4 and in step 5.

Step 7: Repeat the process: construct a circle with center M and passing through point P. Where this new circle intersects segment MN is point R. Construct a square MPQR.

Step 8: Construct arcs centered at the corners of the squares, and inscribed in the consecutive squares as shown. This is not a true spiral, as it is composed of quarter-circles, but it is still called The Golden Spiral. The ancient Greeks believed that the ratio of the sides of this rectangle were the most beautiful of ratios, and they called this The Golden Rectangle.

Many people call this a "spiral staircase":

http://www.yorkspiralstair.com/

It is not really a spiral, though, because a spiral is a "plane curve". This means a spiral is "flat", lying in a flat plane. The staircase above is actually a 3-dimensional curve called a "helix". You may have heard that the DNA molecule is in the shape of a helix.

The Cardioid

The cardioid is the envelope of circles with their centers on a given circle, and passing through one point on the given circle. The word Cardioid comes from the Greek word for heart, and it is a heart-shaped curve. The caardioid can be the basis for many beautiful geometric designs, such as the one below, constructed by a geometry student.

You will find a step-by-step set of instructions for constructing a cardioid below:

1) Construct a circle using a compass. Construct a horizontal line through the center of the circle, using a ruler. Construct a line through the center perpendicular to the horizontal line. Point P is the point that will begin our envelope. Now bisect the right angle formed by these two perpendiculars. Continue to construct angle bisectors until your circle is divided into 16, 32 or 64 equal parts. The more divisions on the circle, the more complex your design will be.

2) Construct a circle using one of the division marks on the given circle as the center, and point P as a point on the circle. Continue this process to create more circles, passing through point P.

You can color your construction in many different ways, to create beautiful designs!

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