- How To Construct Square Root Spiral With Compass Problems
- How To Construct Square Root Spiral With Compass
- How To Construct Square Root Spiral With Compass Worksheet
You can construct the square root of any positive integer with a ruler and compass. I believe you only need a square and compass to construct this. Construct the midpoint M of segment AC, construct the circle with center M passing through A, construct the line perpendicular to AB through B, and let D be one of the intersections of that line with the circle centered at M (call the other intersection E).
Phi is most often calculated using by taking the square root of 5 plus 1 and divided the sum by 2:
√5 + 1
This mathematical expression can be expressed geometrically as shown below:
Three circle construction:
Put three circles with a diameter of 1 (AB and DE) side by side and construct a triangle that connects the bottoms of the outside circles (AC) and the top and bottom of the outside circles (BC). The dimensions are as follows:
AB = 1
AC = 2
BC = √5
DE = 1
The line BC thus expresses the following embedded phi relationships:
BE = DC = (√5-1)/2+1 = (√5+1)/2 = 1.618 … = Phi
BD = EC = (√5-1)/2 = 0.618… = phi
This simple and elegant way of expressing the most standard mathematical expression of Phi was discovered and contributed by Bengt Erik Erlandsen on 1/11/2006.
Construction by compass and ruler:
Scott Beach developed a way to represent this calculation of phi in a geometric construction:
How To Construct Square Root Spiral With Compass Problems
As Scott shares on his web site:
Triangle ABC is a right triangle, where the measure of angle BAC is 90 degrees. The length of side AB is 1 and the length of side AC is 2. The Pythagorean theorem can be used to determine that the length of side BC is the square root of 5. Side BC can be extended by 1 unit of length to establish point D. Line segment DC can then be bisected (divided by 2) to establish point E.
The length of line segment EC is equal to Phi (1.618 …).
The length of line segment DB is 1 and the length of line segment BE is the reciprocal of Phi (0.6180…).
Taken together, DB and BE constitute a graphic representation of the Golden Ratio.
Constructing the Golden Rectangle With respect to the Golden Ratio
by Leanne May
by Leanne May
The ratio, called the Golden Ratio, is the ratio of the length to the width of what is said to be one of the most aesthetically pleasing rectangular shapes. This rectangle, called the Golden Rectangle, appears in nature and is used by humans in both art and architecture. The Golden Ratio can be noticed in the way trees grow, in the proportions of both human and animal bodies, and in the frequency of rabbit births. The ratio is close to 1.618. Whoever first discovered these intriguing manifestations of geometry in nature must have been very excited about the discovery.
A study of the Golden Ratio provides an intereting setting for enrichement activities for older students. Ideas involved are: ratio, similarity, sequences, constructions, and other concepts of algebra and goemetry.
Finding the Golden Ratio. Consider a line segment of a length x+1 such that the ratio of the whole line segment x+1 to the longer segment x is the same as the ratio of the line segment, x, to the shorter segment, 1.
The resulting quadratic equation is
A positive root of this equation is
or 1.61803... This irrational number, or its reciprocal
is known as the Golden Ratio, phi .
Now we will construct the Golden Rectangle. First we will construct a square ABCD.
Now we will construct the midpoint E of DC.
Extend DC. With center E and radius EB, draw an arc crossing EC extended at C.
Construct a perpendicular to DF at F.
Extend AB to intersect the perpendicular at G.
AGFD is a Golden Rectangle.
Now we will measure the length and width of the rectangle. Then we will find the ratio of the length to the width. This should be close to the Golden Ratio (approximately 1.618).
Now we will take our Golden Rectangle and continue to divide it into other Golden Rectangles.
Within this one large Golden Rectangle there are six other Golden Rectangles.
When you measure each Golden Rectangles length and width you will see that the ratio of the length to the width is the Golden Ratio (spproximately 1.618).
Now we will construct the spiral through the whole Golden Rectangle.
How To Construct Square Root Spiral With Compass