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Geometry (What the acorn said when he grew up) (7/00)

Ok, I don't have room or time to write a whole geometry book, but I would like to

include a couple of my favorite topics in this 'area' (heh).

 

• Basic Definitions | Regular Polygons
• Tessellations | Polyhedra
• Conic Sections
• Perimeter, Area, Volume

 

(c) 1997-2007 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.

(Back to Other Stuff: [ Statistics | Number Theory ] . . . More subjects on the way. Suggest one!)



 

In the beginning . . . (top of page)

Euclid (about 2500 years ago, writing his famous 'Elements') started his definitions with,

"A point is that which has no part."

You could say the same about my hair, but he makes 'a good point'; he meant that a point

is an idealized mathematical concept, not a sloppy dot on a piece of paper. He then went on

to define lines, circles, polygons, and came up with Euclid's Five Postulates, dealing with

betweenness and uniqueness of circles and parallel lines and such. If you drop the fifth, you

enter the evil world of "Non- Euclidean" geometry, where you get zero (Riemannian) or an

infinite number (hyperbolic) of lines thru a point 'parallel' to a given line.

 

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Lines, Rays, and Polygons (top of page)

A line has infinite length, but a segment has length equal to the distance between its two endpoints.

A ray starts at one point and goes on and on forever (like that Madonna song).

A polygon consists of a series of segments proceeding from point to point; the popular idea of

a polygon is that it's 'simple' (no self-intersections) 'closed' (a loop, not a path), and usually 'convex'

(no scooped out parts). Let's count the sides:

triangle (3), quadrilateral (4), pentagon, hexagon, heptagon, octagon, nonagon, decagon, . . .

A polygon is regular if all its sides are the same length and all its interior angles are equal,

like a square or a stop sign (octagon).

Here's a table of the number of sides n of a regular polygon and the interior angle a (degrees)

for each; the formula being: a = 180 * [(n - 2)/n] ;

 

 

n	3	4	5	6	7	8	9	10	12
a	60	90	108	120	128 4/7	135	140	144	150

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Tessellations - by popular demand! (7/00) (top of page)

Often called tilings or mosaics, tessellations are (possibly repeating) patterns of polygons that

fill the entire plane. The most common one is like a chessboard, or graph paper, or a screen door.

 

Two types of repeating tessellations made of regular polygons are regular and semi-regular.

Regular tessellations consist of all the same type of polygon, like all squares, and semi-regular

ones have more than one type, like hexagons and triangles. Both types are required to have

the same set of polygons (in the same order) at all corners.

 

The key is to look at the angles of the polygons at a corner; if they don't add up to 360 degrees,

then the tessellation won't work. I like the notation <6, 6, 6> for the tiling of all hexagons, like many

bathroom floors or beehives, because each vertex (corner) has three 6-sided polygons meeting there.

 

The number of regular tessellations is very limited as seen in the table above; only 60, 90, and 120

are divisors of 360, giving the three possible patterns below.

 

 

The Regular Tessellations

< 3, 3, 3, 3, 3, 3 >  < 4, 4, 4, 4 > < 6, 6, 6 >

 

The world of semi-regular tilings is larger; there are at least nine of them, from the simple to

the complex, found in the table below. Check that they all add up to 360, such as < 4, 8, 8 >

(my kitchen floor) which gives 90 + 135 + 135 = 360. Are there any more? You decide!

Note: Although < 5, 5, 10 > adds up correctly, it's physically impossible to continue the pattern for very long.

 

 

Semi-regular Tessellations

< 3, 3, 3, 3, 6 >   Does this exist??   < 3, 3, 6, 6 > < 3, 6, 3, 6 > < 3, 3, 3, 4, 4 > < 3, 3, 4, 3, 4 > < 3, 4, 6, 4 > < 4, 8, 8 > < 3, 12, 12 > < 4, 6, 12 >

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Polyhedra (top of page)

These are like tessellations that don't quite make it all the way around, and have to be 'bent up' into the

third dimension. For example, three squares at a corner would form a cube and be called < 4, 4, 4 >.

 

When drawn on paper, the pattern would leave a gap of 90 deg, and this gap actually determines how

many corners (vertices) the shape will have. Using the Euler characteristic of 2 for a sphere (V - E + F = 2),

we can prove that the total gap must be 720 deg, so a cube has to have 720/90 = 8 total vertices. It does.

 

 

The Five Regular Polyhedra ("Platonic Solids")

Tetrahedron < 3, 3, 3 > 4 triangles Octahedron < 3, 3, 3, 3 > 8 triangles Hexahedron (cube) < 4, 4, 4 > 6 squares Icosahedron < 3, 3, 3, 3, 3 > 20 triangles Dodecahedron < 5, 5, 5 > 12 pentagons

You can make your own regular and semi-regular polyhedra by cutting out dozens of cardboard

polygons and using scotch tape for the edges. I used to sell these things as 'geometric art' when I

was a kid. "Get your truncated icosahedron over here, one dollar!" (What sport uses this shape?)

 

The Thirteen Semi-Regular Polyhedra ("Archimedean Solids") Graphics produced with Mathematica

Truncated Tetrahedron < 3, 6, 6 > 4 triangles 4 hexagons Truncated Cube < 3, 3, 3, 3 > 8 triangles 6 octagons Cubocta- hedron < 4, 4, 4 > 8 triangles 6 squares Truncated Octahedron < 5, 5, 5 > 6 squares 8 hexagons Snub Cube < 3, 3, 3, 3, 4 > 32 triangles 6 squares      It's a fun puzzle to figure out, for each solid, how many: .vertices (V) (corners) .edges (E) (edges) and .faces (F) (polygons).     Icosi- dodecahedron < 3, 5, 3, 5 > 20 triangles 12 pentagons Truncated Cuboctahedron < 4, 6, 8 > 12 sqrs, 8 hexs, 6 octagons Rhombicub- octahedron < 3, 4, 4, 4 > 8 triangles 18 squares (Thanks Colin!) Truncated Dodecahedron < 3, 10, 10 > 20 triangles 12 decagons     Make up variables for the number of each polygon. Follow Euler's rule V - E + F = 2. Then solve for V, E, F ; I dare you!     Truncated Icosahedron < 5, 6, 6 > 12 pentagons 20 hexagons Truncated Icosi- dodecahedron < 4, 6, 10 > 30 sq, 20 hex, 12 decagons Snub Dodecahedron < 3, 3, 3, 3, 5 > 80 triangles, 12 pentagons Rhombicosa- dodecahedron < 3, 4, 5, 4 > 20 tri, 30 sq, 12 pentagons

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Conic Sections (5/03) (top of page)

Mathematicians think of a cone as a line spun around an axis at an angle.

This creates a "double cone" --- two cones point-to-point, touching at the vertex.

You can slice the cone at various angles and get different shapes; these curves

are called conic sections for that reason.

If the plane is perpendicular to the axis you get a familiar shape: a circle.

If the plane is slightly tilted, you get a stretched-out circle, called an ellipse.

Make the plane parallel to a side of the cone and the shape becomes a parabola.

And if the plane slices both halves of the cone, you get a two-branched hyperbola.

 

 

Dan's favorite proof from high-school geometry: . . . . . . . the "baseball & basketball proof" of the ellipse The ellipse can be defined as the set of points P such that the sum of the distances PF1 and PF2 is a constant. The F1 and F2 are called the foci (focuses) of the ellipse. In the picture to the right, the intersection of the cone and plane forms a curve, a typical point on it is P. Can we prove this curve is an ellipse by the new definition?   We form two spheres tangent to the inside of the cone and both touching the plane, one from above touching at F1 and one below touching at F2. The two tangent segments, PF1 and PA, from P to the upper sphere are equal (to d1), and the segments PF2 and PB tangent to the lower sphere are equal (to d2). (Yes, I drew the picture.)   So we get : d1 + d2 = AP + PB = AB = constant!

 

 

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Perimeter, Area, and Volume Formulas (or is it 'formulae') (3/01) (top of page)

1. The perimeter of a polygon (or any closed curve) is the distance around.

2. The area of a simple, closed, planar curve is the amount of space inside.

3. The volume of a solid 3D shape is the amount of space displaced by it.

 

 

There are plenty of good formulas for figuring these out: the answers have one,

two, or three dimensions; linear units, square units, or cubic units.

 

1. Perimeter formulas: P = 4s (square) , P = 2L + 2W (rectangle) , P = a + b + c (triangle) ,

. . . P = C = 2 pi r = pi D (circle) Also c =(a^2 + b^2) (right triangle; see trig page)

 

 

2. Area formulas: A = s^2 (square) , A = LW (rectangle) , A = (1/2) b h (triangle) ,

. . . A = [s(s - a)(s - b)(s - c)] (triangle; s = (a+b+c)/2) , A = b h (trapezoid) ,

. . . A = pi r^2 (circle)

 

 

3. Volume formulas: V = s^3 (cube) , V = LWH (rectang. box) , V = A h (any cylinder) ,

. . . (A = area of base) , V = (1/3) A h (any pyramid or cone) , V = (4/3) pi r^3 (sphere).

 

 

 

** Pictures coming soon for the formula section! **

 

Well, that's it for now. Check back often for new stuff!

Click below for other topics, or visit the ask dan page!

 

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