In order to save paper, I have not printed this lab. However, if you prefer, you may print the lab yourself.

Answer the following questions on a sheet of paper.

For example, here experts collect data for a large sample of individuals and assigned the participants in one of two classes. Each individual corresponds to a point in an n-dimensional space where n is the number of measurements recorded for each individual. Mathematics is then used to separate the classes via a plane, similar to the idea of linear regression (which we'll see later on this semester), but instead of finding a "best fit" line to all of the data, we find the higher dimensional plane that best separates the data into classes. |

New individuals are then classified and diagnosed by a computer using the separating plane.

**Breast Cancer**
When a tumor is found, it is important to
diagnose whether it is benign or cancerous.
In real-life,
9 attributes were
obtained via needle aspiration of a tumor such as clump thickness,
uniformity of cell size, and uniformity of cell shape.
The Wisconsin Breast Cancer Database used the data of
682 patients whose cancer status was known.
Since 9 attributes were measured, the data was contained in a
space that had 9 physical dimensions.
A separating plane was obtained.
There has been 100% correctness on computer diagnosis of 131 new
(initially unknown) cases, so this method has been very successful.

**Heart Disease**

View the description of the data. Scroll down to number 7, which explains the attributes used in the analysis. If you wish, you can look at the abbreviations and then scroll down to identify the meaning via the complete attribute documentation descriptions.

This data was the real data that was used to find a separating plane in this higher dimensional data space. New patients have since been diagnosed using this plane. You may now quit Excel without saving your file.

Similarly, we can use indirect ways of trying to gain an appreciation for the counterintuitive behavior of 4D objects. A hypercube is one of the "easiest" 4D objects to try and understand. Yet, physicists and mathematicians assert that it cannot be the shape of our universe since it has an edge, which means that there would have to be something on the other side of the edge.

First look at Dimensional Connections and Into the Fourth Dimension

Next examine Counting Boundaries, Hypercube Boundary, and the movie The Eight Cubical Faces of a Hypercube - to play the movie, use the .

Now examine A Cube Unfolded and A Hypercube Unfolded.

Explore the movies Rotating a Cube and Rotating a Hypercube by using the . If it is viewed properly, the second movie should stretch the limits of your imagination.

Artist Toni Robbin, from the

He also mentions that he finds it a privilege to live in a time where advanced mathematics is represented in pictures instead of only equations.

Some cosmologists expect the universe to be finite, curving back around on itself. Historically, the idea of a finite universe ran into its own obstacle: the apparent need for an edge. Aristotle argued that the universe is finite on the grounds that a boundary was necessary to fix an absolute reference frame, which was important to his worldview. But his critics wondered what happened at the edge. Every edge has another side. So why not redefine the "universe" to include that other side? German mathematician Georg Riemann solved the riddle in the mid-19th century. As a model for the cosmos, he proposed the hypersphere--the three-dimensional surface of a four-dimensional ball, just as an ordinary sphere is the two-dimensional surface of a three-dimensional ball. It was the first example of a space that is finite yet has no problematic boundary. One might still ask what is outside the universe. But this question supposes that the ultimate physical reality must be a Euclidean space of some dimension. That is, it presumes that if space is a hypersphere, then that hypersphere must sit in a four-dimensional Euclidean space, allowing us to view it from the outside. Nature, however, need not cling to this notion. It would be perfectly acceptable for the universe to be a hypersphere and not be embedded in any higher-dimensional space. Such an object may be difficult to visualize, because we are used to viewing shapes from the outside. But there need not be an "outside."

For tomorrow, go through the study guide for the test and message me or write down at least one question or comment for a review session during class. Recall that the advice from last semester's students was to know everything on the study guide. In addition take a try of ASULearn Material Review Quiz [participation requires at least one try of the quiz, but the specific grade does not matter.]

We will go over the lab in class tomorrow. If you are finished, you may leave, or stay to work on homework or ask me questions.