Nov 19, 2018 Let’s see how to construct this spiral. We start with an isosceles right-angled triangle. The length of the equal sides is 1 unit and the length of the hypotenuse is √2 units. Now taking the hypotenuse of the right-angled triangle as one of the sides and the length of the other side as 1 unit, construct another right-angled triangle as shown in the ﬁgure. 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.
How To Construct Square Root Spiral With Compass Protractor
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.
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Question Corner and Discussion AreaAsked by several students on August 14, 1997:
I would like to know the three ancient impossible constructions problemsusing only a compass and a straight edge of Euclidean Geometry.The three problems are:
- Trisecting an angle (dividing a given angle into threeequal angles),
- Squaring a circle (constructing a square with the samearea as a given circle), and
- Doubling a cube (constructing a cube with twice the volumeof a given cube).
People tried for centuries to find such constructions. It was not until thedevelopment of 'abstract algebra' in the nineteenth century that itwas proven these constructions were impossible.
For those of you who are interested, the basic idea of the proofis as follows.
Since the only shapes you can draw with a compass and straightedge areline segments and circles (or parts of circles), the only ways ofconstructing new points out of old points to to take the intersection oftwo lines, two circles, or a line and a circle. Now, if you write downthe general equations for these intersections and try to solve for one of thecoordinates of the intersection point, you will end up with either a linearor quadratic equation. The coefficients of the equation involve thecoordinates of the old points (or various sums, differences, products,or quotients of them).
Linear equations can be solved by simple division: the equationax = b has as its solution x = b/a. Quadratic equations can besolved using the quadratic formula.In each case,we see that the only arithmetic operations required tocalculate the new coordinatefrom the old coordinates are addition, subtraction, multiplication,division, and taking square roots.
Therefore, if you start with some initial points whose coordinatesare all rational numbers, then apply any sequence ofcompass-and-straightedge construction techniques, the coordinates ofthe points you end up with will be a very special kind of number: theywill be obtainable from the rational numbers by a sequence of operationsinvolving only addition, subtraction, multiplication, division, and theextraction of square roots.
The reason the three classical constructions are impossible is that theyare asking you to be able to construct points whose coordinates are notnumbers of this type.
Square Root Calculator
Proving that they are not numbers of this type requires some very advancedmathematics from an area called Field Theory. I'll sketch some ofthe essential ideas.
The key theorem is this:
If x is a number obtainable from the rationalsusing only addition, subtraction, multiplication, division, and the takingof square roots, then x is a solution to some polynomial equationwith rational coefficients. Moreover, if one factors out irrelevant factorsfrom this equation until one gets down to an 'irreducible' polynomialequation (one that can't be factored any further and stillhave rational coefficients), the degree of this polynomial will always bea power of 2.Intuitively speaking: each time you take a square root, you usuallydouble the degree of the polynomial required to represent the number.For example, can be represented as a solution to thequadratic equation , whose degree is 2. (Of course,it's also a solution tothe cubic equation whose degree is 3, not a power of 2;but that's because this equation can be factored, and after factoring outthe redundant factor of x you are left with the quadratic, which is irreducible: it can't be factored any furtherand still have rational coefficients). If we take a square root a secondtime, getting the number , it's a root of thefourth-degreepolynomial (which is irreducible).A number like ,also obtained by twice taking a square root, is also theroot of an irreducible fourth-degree polynomial:. A number obtained by taking a square rootthree times, like , is typically the root of an irreducible8th degree polynomial, like .
This intuitive understanding is not always correct, though. For example, + , even though it is obtained using twosquare roots, is the same as , which involves only onesquare root. So the intuitive argument only serves to show you that theclaims of the theorem are reasonable. Actually proving the theorem(and proving, not just that there's some irreducible polynomialequation for x whose degree is a power of 2, but thatevery irreducible polynomial equation for x also has that samedegree) involves many advanced ideas.
Using the theorem, it is easy to prove the impossibility of the threeconstructions:
Doubling a cube is impossible because if you start with a cube of sidelength 1, you would need to cunstruct a cube whose side length is thecubed root of 2. But the the cubed rootof 2 is a solution to the irreducible equation whosedegree, 3, is not a power of 2.
Squaring a circle is impossible because if you start with a circle ofradius 1 you would need to construct a square whose side length is. But this is a so-called 'transcendental number': it isnot the solution to any polynomial equation with rationalcoefficients, let alone one whose degree is a power of 2.
Trisecting an angle is impossible because if you start with an angleof 60 degrees (which is easily constructible), you would then need tobe able to construct an angle of 20 degrees. This would be equivalentto constructing a point whose coordinates are the cosine and sine of20 degrees. This is impossible because cos(20 degrees) is asolution to the irreduciblepolynomial equation whose degree, 3,is not a power of 2.
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