These ideas will take a while to sink in so we will reinforce the material with different activities. From the video, you should take away knowledge about Jeff Weeks' mathematical style of doing research along with what he works on, and what he thinks about mathematics.

Data is collected for a large sample of individuals where individuals have been assigned to one of two classes by experts. 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, but instead of finding a "best fit" line to all of the data, we find the 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 - Be sure that you have read the text above before
performing this activity.**

Find a partner. One of you should read this page as the other follows the directions. View the real-life numerical data that was actually used in the heart disease analysis. Using Select All and then Copy under Edit, copy the numerical data from this link. Open up Word and paste the data into Word. Under Edit, Replace all of the instances of , with ^t . Then under Edit, Select All and then Copy. Open up Excel and paste the data into Excel. It may take a while since there is a lot of data. Each column is a different dimensions worth of data. How many dimensions is this space? Each patient is a different row. How many patients were studied?

View the description of the data. Scroll down to number 7. Use this to identify exactly which attributes were used in the analysis by looking at their abbreviations and then scrolling 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. I wanted you to see how you can place it into Excel and to have some experience the actual data that was used. It is not often that one gets the chance to do this, because people rarely make their data sets available to others. 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.

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."