NUMB3RS SEASON 4 EPISODE HOW TO

Using matrices to describe rotations is less intuitive than describing rotations by their axis, but it makes the job of determining what happens when several rotations are applied to an object almost trivial to compute. The composition of those two rotations is again a rotation, so what is its axis of rotation and by how many degrees is the apple rotated? The answer is not easy to come up with, but it is possible to learn the answer through inspection. The former description of angles rotated around a given line is a fairly straightforward description and has the advantage that it is easy to visualize, but it's drawback is that it is not easy to compose rotations - imagine you have an apple, and you rotate that apple 60 degrees about a vertical axis going through its center of gravity and then you rotate 60 degrees around a horizontal axis through the center of gravity. (Another popular and useful way to describe rotations is with Quaternions, but we will not discuss those here.) Two stand out as being valuable to our present study: a description of the amount of rotation around a named axis and with a matrix. Rotations can be described a few different ways in mathematics. They both have one 'hole', and one can be continuously deformed into another.ДЄ coffee cup or a donut? Johann Listing can't tell the difference! A classic joke about this is that a topologist cannot distinguish his morning donut from his morning cup of coffee.

NUMB3RS SEASON 4 EPISODE HOW TO

More specifically, topologists are interested in how to describe objects, when you are allowed to continuously deform them. Listing worked in a field of mathematics known as topology and even coined the moniker topology to describe the field, which examines the shape and orientation of objects. Listing's Law was developed by the German mathematician Johann Benedict Listing who lived from 1808 to 1882. We will describe this example in more detail after we've built up the mathematical framework to describe rotations.

This formulation cannot be the case because it would imply that the eye can travel a path and end looking the same direction as when it started, except being rotated. Listing's law is sometimes mistakenly stated as saying that when the eye moves from any sight direction to any other sight direction, the axis that it is rotated on is the unique one that is perpendicular to both the initial and final lines of sight. No matter what path the eye takes to get to a given direction of sight, the image seen will always be the same - there will be no rotation.ДЄnother formulation of his law states that the eye, when moving from looking straight ahead to another direction, then the axis of rotation is the unique line which intersects and is perpendicular to both the initial and final lines of sight. One interesting consequence of Listing's Law is that whenever a person is looking in a particular direction, the eye is always oriented in the same direction. Listing's law states that all eye rotations have an axis of rotations which is contained within the same plane, called Listing's Plane. In this episode, Charlie and Amita use Listing's Law to predict the path that a criminal takes through a vault.

Numb3rs Season 4 Episode 5: Robin Hood Numb3rs Season 4 Episode 5: Robin Hood

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