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**geometry** **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).
- (c) 1997-2001 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.
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**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 (no whiskey jokes!) 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.

**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**

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**

**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**tofigure 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**

**Rhombicosa-****cuboctahedron****< 3, 4, 4, 4 >****8 triangles****18 squares**

**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**

**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!****[ statistics | number theory | geometry | top of page | lessons index]**+ Basic Skills + Arithmetic, Prealgebra, Beginning Algebra + Precalculus + Intermed.Algebra, Trigonometry, Functions & graphs + Calculus + Limits, Differential Calc, Integral Calc, Vector Calc + Beyond Calculus + Linear Algebra, Diff Equations, Math Major stuff + Other Stuff + **You are here**1. Statistics, Number Theory 2. **Geometry**Other requests?**[ home | info | meet dan | ask dan | dvc ]**This site maintained by B & L Web Design, a division of B & L Math Enterprises.