The Golden Mean - The Golden Section
The Phi Ratio - The Life Ratio
Geometric / Mathematical Proof
A scroll-down lesson from Brian Joseph
Step One
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Begin with a square. The actual size of the square is irrelevant so long as each side is the same length (definition of a square). To keep the math as simple as possible, we'll arbitrarily assign a value of one (1) as the length of each side. If you don't enjoy math, refer to the Rectangle Construction page. |
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Step Two
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This exercise is to make four equal circles,
each centered at a corner.
Open your compass a little over half as wide as a side of the square. Just set the pivot at a corner and eye it. Using light pressure (you will erase these marks later) spin a circle around a corner of the square. Do this for each of the four corners without changing the open-size of the compass. |
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Step Three
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This exercise is to vertically divide the square
into two equal sections.
"The shortest distance between two points is a straight line." Use your ruler to draw a straight line through points where the circles cross on the up/down axis. |
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Step Four
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You now have a square with lengths of 1 and (1/2)+(1/2).
Use your ruler to draw a straight line from the center of the bottom side of the square to the upper right-hand corner. |
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Step Five
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Now use your ruler to extend the lower side of the square out another length of 1 (just eye it). |
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Step Six
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Set your compass pivot at the center of the lower side of the square. Open the compass to the top right corner of the square and swing down to the extended lower line you drew in the previous step. It is not neccessary to complete as much of the circle as I did. |
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Step Seven
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For a right triangle (a triangle that has one
90 degree angle; angle <ab>
in diagram), the length <c>
opposite the right angle <ab>
can be determined with the formula a2+b2=c2
(Pythagorean Theorem).
We know that angle <ab>=90 degrees (because we began with a square). |
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Step Eight
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We know that b=1 because that is what we chose
for the length of a side of the square. We know that a=1/2 or .5
because it is half the length of that side. All we have to do is
plug in the numbers to the formula.
a2+b2=c2 a=1/2, a2=(1/2)2=1/4 or a2=(1/2)*(1/2)=1/4 b=1, b2=12=1 or b2=1*1=1 c2=a2+b2 or c2=1/4+1 |
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Step Nine
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If c2=5/4, then <c> is the square root of 5 divided by two (the square root of four). |
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Step Ten
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The square root of five is 2.2360679... (the
three dots means the decimals go on forever, like pi does: 3.1419...)
Divide it by two to get c=1.1180339... |
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Step Eleven
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We know that c=1.1180339 approximately.
Now we get to find out the length of <A>,which
is the whole line BC.
<A> is the length BC, and B=1. We know that the length from the center of line <B> to the end of line <C> is equal to length <c> or ~1.1180339 because we used our compass to mark the endpoint of line C (we brought line <c> clockwise onto the extension of line <B>). Since <A> is just 1/2 (or .5) longer than <c>, then A=.5+1.1180339... or A=~1.6180339 |
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Step Twelve
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What we've figured out:
A=B+C A=1.6180339... B=1 C=?? c=1.1180339... There's two ways to find the length of <C>. First, <C> is equal to length <c> minus half of <B>, or c=0.6180339... The other way is to subtract <B> from the whole (<A>): C=A-B (C=A-1), or C=0.6180339... |
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Step Thirteen
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Welcome to the Golden Mean.
The magic of this length is that when you divide the whole length by its larger piece, you get the same ratio as you do if you divide the larger of the two pieces by the smaller. <A> divided by <B> is equal to <B> divided by <C>. 1.6180339.../1=1/0.6180339 1.6180339...=1.6180339... |
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| You have completed the proof. Continue below to plug in |
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Step Eight-a
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We'll use the same square and give each of its
sides a length of 6 instead of 1. Therefore,
b=6. We know that a=(1/2)b or 3 because it is half the length of
that side. All we have to do is plug in these new numbers to the
formula.
a2+b2=c2 a=3, a2=32=9 or a2=3*3=9 b=6, b2=62=36 or b2=6*6=36 c2=a2+b2 or c2=9+36 |
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Step Nine-a
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If c2=45, then <c> is the square root of 45. |
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Step Ten-a
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The square root of 45 is 6.7082039... (use a calculator...) |
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Step Eleven-a
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We know that c=6.7082039 approximately.
Now we get to find out the length of <A>,which is
the whole line BC.
<A> is the length BC, and B=6. We know that the length from the center of line <B> to the end of line <C> is equal to length <c> or ~6.7082039 because we used our compass to mark the endpoint of line <C> (we brought line <c> clockwise onto the extension of line <B>). Since <A> is just 3 longer than <c>, then A=3+6.7082039 or A=~9.7082039 |
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Step Twelve-a
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What we've figured out:
A=B+C A=9.7082039 B=6 C=?? c=6.7082039 There's two ways to find the length of <C>. First, <C> is equal to length <c> minus half of <B>, or c=3.7082039 The other way is to subtract <B> from the whole (<A>): C=A-B (C=A-6), or C=3.7082039 |
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Step Thirteen-a
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Welcome back to the Golden Mean.
The magic ratio (where A=B+C) of A/B=B/C never changes, regardless of the size of the original square. Now that you understand the math, you can test this with any number you choose. <A> divided by <B> is equal to <B> divided by <C>. 9.7082039/6=6/3.7082039 1.6180339...=1.6180339... |
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| links at right. |
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