N-Space:Boy BandErik ScottHighline Community College

Doing math 22/7?!!

Are you nuts?

Math as Puzzles (CMRL Math)Try to connect the two dots below using a combination of red and blue arrows. Each red arrow means add six and each blue arrow means subtract fifteen. Can it be done?411

Geometry and the Structure of the UniverseGeometry in Chemistry:Geometry in Physics:

The Magical Symbols of MathematicsIn Algebra:In Calculus:In the Sciences:

They can even be used as a signature:

Comparing dimensions visually:A two-dimensional (2-D) image:A three-dimensional (3-D) image:

What is a dimension?It is a category we use to describe something.For objects: Height or WidthFor people: Honest or Fun

How does a person decide if something is 1-dimensional, 2-dimensional, etc?Think of the different adjectives you could use to completely describe the quantity, object, or person.Group the adjectives into categories.Eliminate redundant categories to find the smallest number of categories required.Count the number of categories to obtain the overall dimension of the object.

Descriptions of the categories weve found:Category 1: My placement left to right.Category 2: My placement front to back.Category 3: My placement up and down.Notice that all three descriptions are completely distinct adjectives from categories 1 and 2 are not enough to describe category 3.

Mathematical Equivalents(Linear) Independence This is our requirement that each category is not describable by a combination of descriptions from other categories.Basis A set of categories with the two properties listed above. Its the smallest, most economical collection of categories.Spanning SetThis is our requirement that the collection of categories completely describe the situation of interest.

Cloud Nine,

or

Just a Nine-Dimensional Subspace?

Personals Ads:

The span of one vector:Marital Status

The span of two vectors:Marital Status and Race/Ethnicity

The span of three vectors:Marital Status, Race/Ethnicity, and Gender

The space searched by Match.com may be 14-Dimensional!!

Spaces of Interest to MathematiciansInfinite-dimensional spaces:This is a set of functions with basis1, x, x2, x3, .that includes functions such as

The Klein Bottleand its familiar immersion into R3:

The Klein Bottle in 4-D:

Welcome people to my talk.[Slide]Explain my intent to answer the questionsWhat are dimensions and higher-dimensional spaces? andHow could a person get excited about it in the first place?Mention that I want to start with the latter, since a lot of people have this question about doing any math; it often takes the form:Well, maybe they dont phrase it with a math pun, but you get the idea.How did I get hooked? It started in elementary school. The math I was taught usually took the form of puzzles and games. I couldnt do everything, but had enough fun to stay interested.Science always fascinated me. I asked a lot of questions about where things come from, and how they work.In high school, I began to see how scientists talk about the world, and started seeing important connections with math.[Slide]Math began to take on an almost magical significance: a shape I drew on a piece of paper could match the structure of diamond [Slide] or be the same as the motion of planets.Meanwhile, my math and science classes were introducing me to exotic, and sometimes elegant, symbols for describing phenomena.[Slide]Sigma notation offered a way to add an infinite number of numbers, and gave a mathematical interpretation of Zenos paradox.[Slide]The integral symbol fascinated me to no end. I pressed my precalculus instructor to explain what it was for when I found it lurking at the back of our textbook. That it could be used to find areas of complicated shapes was icing on the cake.[Slide]Ultimately, I met the Schroedinger equation. Talk about magic! This string of symbols on a piece of paper describes the behavior of electrons in atoms; it gets at the very essence of our universe!

Of course, there was always time for fun and games with this stuff, too.The source of my passion for mathematics the magical process of being able to start with a set of symbols and wind up manipulating the universe itself, such as creating a star in a laboratory was captured perfectly in a segment of video I first saw about four years ago. It appeared in a game called Final Fantasy VII (ironic title) during the cataclysmic final battle. What you will see is part of a magical spell cast by the antagonist, Sephiroth.[Advance Slide while video loads][Pause here through video.]As you can see, the segment puts the connection between mathematics, science, and the operation of the universe at its fore. Add classical music, lyrics from Carmina Burana, and pack it in a video game and Im in 7th heaven.[Slide]I think the video segment also successfully leads us back to the mathematical topic of my talk, namely dimensions. Computer games in particular confuse the issue considerably. Nearly all of the new games are 3-D of some sort. But the fact is, theyre displayed on a SCREEN! And a screen is 2-D. Nevertheless, there is a difference, as you can see in these pictures.Soooooo, what is a dimension? (Ask for examples -- personal curiosity.)My answer to this question is...Use the phrases: What is the pictures dimensions? and Passion is a dimension of my personality.Now that we can think about dimensions as categories, lets see if we can make the difference between 2-D and 3-D quite clear.[Slide]Talk through the steps, then comment on the fact that usually only quantifiable categories are allowed.

Task: Lets determine how many dimensions are required to describe my position in the room. Think about trying to tell your friend who didnt attend where I was in the room. Move to a spot in the room: Where am I?What categories arose among things people said? Move to a new spot, exactly left or right of where I was.In what direction did I just move? (Sideways) Move to a new spot, exactly forward of where I was.Where did I move now? (Forward) Move atop a chair or table?Same question. (Up)Are there any other categories people can think of to describe my position?To summarize, we found these categories.[Slide x3]This seems to suggest that my location is defined by three dimensions. But we do have to check whether we can be more efficient in describing my location.[Slide]A quick examination says we cannot (read slide).Read off.

So you now have some sense of what a dimension is, and even how mathematicians talk about them. Now we can address the REALLY important question.How do higher-dimensional spaces affect your love life?Heres an example of the modern approach to personals ads. Notice that the criteria are independent. Do they span?

What is the traditional beginning to a personals ad? Isnt it usually a string of three or so letters, like SBF, for Divorced, Black, Female?

Are the categories each descriptor comes from all independent? Then this collection describes a three-dimensional space.We start with marital status, which has three variations.Add to it four variations in race or ethnicity.And conclude with two variations in gender. That leaves us with a 3-D vector space in which to place ourselves.This has up to 14 independent categories. One could argue that Income and Occupation are not entirely independent, though.Mention connection of the smooth functions with convergent power series to Hilbert Space and quantum mechanics.

QM is a godsend for mathematicians - if anyone asks, Where is that used? we can almost always rely on QM and general relativity to make use of even the most obscure mathematical ideas. These functions dont even require that abstract a description. To do calculations on computer we can reduce certain operations to matrix calculations .What dimension is the Klein bottle? Is it 2-D, 3-D, or 4-D? Usually what is said is, Heres the Klein bottle, except that it doesnt really intersect itself. Thats because it really lives in four dimensions. What exactly does that mean?[Explain how to interpret oriented rectangle for torus, reference Asteroids, then point out relative positions of points the fact that a point along the equator will wind up next to points along the green edges when immersed into 3-D. Show Robs animation to reveal need for 4-D.]Animation Rob Grube, 2000