## Thursday, June 16, 2005

### Polygons...!!!

When was the last time that you wanted to draw an Eleven sided Polygon,
But were Stymied by your inexperience with The Fundamentals of Geometry...??? ( ! )
It's easy enough, with only a few simple tools:
Such as;
A Sheet of Paper or Other Inscribable Surface,
A StraightEdge; ( Commonly called a Ruler ),
A Compass ( The Kind that Allows You to Draw
a Circle or Part of a Circle ( An Arc! ),
A Protractor; ( Typically--A Half-Circle of Plastic
with Marks to Indicate Angular Progress along The Arc )
and a Writing or Inscribing Utensil.

To Draw an Eleven Sided Polygon ( An Ennagon )
Simply divide The Number of Degrees in a Circle; ( 360 )
By The Number of Desired Sides ( 11 )--
Which Equals 32.7273!

Draw a Circle,
Carefully Marking where The Center is,
With a Radius of The Desired Amount,
Then Draw a Line through The Circle,
So that it passes through The Center,
This Line is to be Called 'Alice'.
Where this Line passes Through The Edge of the Circle,
Mark as 'Betty'.
So that it's Center Marking Overlaps The Center of your Circle.
Then Find Where 32.7273 is along The Protractors Arc,
And make a small 'Tick' Mark on your Inscribing Surface.
Use your Straight Edge to Draw a Line ( A Finite Ray )
So that it passes through the Center of your Circle--
The Newly made Tick Mark, and The Edge of the Circle.
Where this Line Intersects the Circle, Label it 'Cindy'.
Carefully place The Pointy Metal Spike of your Compass on Betty,
And move The Adjustable Arm back & forth
Until it's Angle reproduces The Length from Betty to Cindy.
Take care to Preserve this Angle on your Compass...!!!
...As you place The Pointy Metal Spike on Cindy
And Make another tick mark along the Circle's Edge,
Moving Away from Betty.
( You should be Moving Counter-Clockwise,
But if you're not, That's OK too...! )
Repeat this procedure, over and over,
Moving The Pointy Metal Spike at each interval
To the Tick Mark that you've just made,
Until your most recent Tick Mark is where Betty is.
It may be entirely possible
That this most recent Tick Mark will not be exactly where Betty is,
But it should be pretty close...!!!
If it's WAY OFF...!!!
Then you may either decide that it's 'Good Enough' or you may want to Do It Over.
When all the Tick Marks are Done;
---
Wasn't that Simple...?
But what if you Don't have a Protractor...!!!
What if you have a Straight Edge, Compass, Writing Utensil & Inscribing Surface...
But no way to determine how much of an Angle 32.7273 would be...???
---
The General Solution!
A General Solution is one that applies
Not only to what you want to do for a Specific Problem,
But will work on Any Problem of that Kind...!!!
This Solution is Not Completely a General Solution,
Since it will only work for Polygons up to 19 Sides...
But that should be plenty,
Because--
Even if you're VERY Neat & Careful with A Straight Edge, Compass & Writing Utensil...
You really can't make Polygons with more sides than this...
With any Reasonable Degree of Reliability...!!!
---
Step One:
Draw a Circle of a The Desired Radius.

The Radius is from Center to The Circle's Edge.
The Diameter is from The Circle's Side to Side.
When you're Calculating The Circumference of a Circle;
The Formula is 2 times Pi times r^2.
It's 2 times... Pi r squared,
Because Pi is a function of the Circumference, Not the Radius,
Even though all Modern Geometers always think of Circles
by their Radius' as how big they are...
The ancient Greeks, that Discovered pi,
thought about The Size of Circles by their Diameters!
But by the time Modern Geometry came along,

Step Two:
Draw a Line through the Center,
Extending through both Sides, And then some.

Step Three:
Use this 'Look Up Table' to find The Polygon that you're Interested in.
Nearly Each Polygon is provided with 2 Right Triangles,
Such that the Accumulative Angles at The Center of the Circle
Will very nearly match the Desired Angle for Creating your Polygon... ( ? )
To Use this Information:
Find The Desired Polygon
Note The Angle, for your Personal Enrichment
Then make a note of The Two Ratios that Follow
The Triangles Angles are given after that,
Again just for your Personal Enrichment,
Along with the Margin of Error.
What you're really;
And Only Interested in though,
Is The Ratios.
So; As an Example--
For an Ennagon;
The Two Triangles are 7:3 & 6:1
These Triangles produce The Angles 23.2 Degrees & 9.46 Degrees,
Which Add Up to 32.66, Pretty close to the Desired 32.73
All the Ratios are given with Fairly Small Integers,
So that the Triangles that you'll be drawing will be easy to measure... ( ! )
Proceed to Step Four...

Look Up Table for Polygon Drawing...
Name......Angle....Ratios......TriAngles........Error
Triangle..120......5:9 + 3:5...60.94 + 59.04....0184
Square.....90......1:1 + 1:1...45....+ 45.......0
Pentagon...72......5:6 + 5:2...50.19 + 21.80....0042
Hexagon....60......5:3 + 9:5...30.96 + 29.05....0184
Septagon...51.43...7:5 + 7:2...35.54 + 15.95....0545
Octagon....45......1:1.........45...............0
Nonagon....40......3:2 + 9:1...33.69 + 6.34.....0303
Decagon....36......2:1 + 6:1...26.57 + 9.46.....0274
Ennagon....32.73...7:3 + 6:1...23.2. + 9.46.....0664
12agon.....30......7:2 + 4:1...15.95 + 14.04....0184
13agon.....27.69...8:3 + 8:1...20.56 + 7.13.....0113
14agon.....25.71...3:1 + 8:1...18.43 + 7.13.....1543
15agon.....24......9:4.........23.96............5625
16agon.....22.5....5:1 + 5:1...11.31 + 11.31....1199
17agon.....21.18...8:1 + 4:1....7.13 + 14.04....0152
18agon.....20......9:2 + 8:1...12.53 + 7.13.....3462
19agon.....18.95...6:1 + 6:1....9.46 + 9.46.....0227

Step Four:
This Step Assumes
That you already know how to make a Right Angle
That Intersects with a given Line.
If you don't know how to do this with a Straight Edge & Compass,
The use the Corner of a Sheet of Paper.
Then:
You want to create a Triangle so that its bottom is The Base Line
This Triangle also needs to have Sides that Equal the Ratio Values.
For this Ennagon:
The first Triangle is 7:3...
That's 7 Units along The Bottom, and 3 Units Up.
These 'Units' may be any length,
So long as they agree with one another.
You can even use different sized Units for the Second Triangle...!
( Although... Why would you WANT to...??? )
To get 7 Units,

Think of The Distance from The Center of your Circle
To the Outer Rim as 4 Units,
Then Divide this Length
in Half, and Half again,
Yielding 1 Unit Increments.
The Double that 4 Unit Length Along the Base Line,
So that it's now 8 Units.
Subtract 1 Unit, So that it's 7 Units.
Draw a Right Angle at this Point.
Extend that Height to 3 Units.
Then Complete the Triangle by Connecting That Point to the Center.
This Newly Drawn Line is The New Base Line!
Repeat this Procedure for The Second Triangle. ( 6:1 )
( You've already got your 1 Units, So this should be easy! )
The Two Triangles now create a Accumulative Angle
That passes through your Circle at 32.66 Degrees.
Make a Tick along the Circles Edge
Where The Second Triangles Hypotenuse Intersects it!
( The Hypotenuse is the longest side of The Triangle! )
Use this 'Measurement' with your Compass
To Mark Off the rest of Vertices ( Points ) for your Polygon.
Connect these Vertices with your StraightEdge!