The Cross Product

Posted on Mon 01 October 2018 in Thesis • Tagged with thesis, quaternions, linear algebra

We're nearly ready to give the geometric interpretation of quaternionic multiplication that we've been working toward ever since we first defined the quaternions. Instead of considering the four-dimensional quaternions, we've spent the last few posts considering three-dimensional vectors, specifically discussing a product operation called the dot product that we can …

The Dot Product

Posted on Wed 12 September 2018 in Thesis • Tagged with thesis, quaternions, linear algebra

Before the hiatus, we began discussing the geometry of vectors in three dimensions. At the end of that post, we discussed why it was impossible to define a multiplication operation on \(\mathbb{R}^3\) that also has an inverse (division) operation. However, there are still two useful "product" operations involving …

Geometry of Three-Dimensional Vectors

Posted on Mon 30 April 2018 in Thesis • Tagged with thesis, quaternions, linear algebra

The previous posts in this series discussed two-dimensional vector geometry and how that geometry can be connected to the algebra of complex numbers. The main result is that the multiplication of complex numbers corresponds geometrically to rotations and dilations of the 2-dimensional plane.

As mentioned in an earlier post, this …

Geometry of the Complex Numbers: Rotations

Posted on Fri 23 March 2018 in Thesis • Tagged with thesis, quaternions, complex numbers

In the previous post in this series, we introduced a geometrical interpretation of the complex numbers as vectors in the 2-dimensional plane. That post also explained the geometrical significance of the addition of complex numbers as vector addition. In this post, we'll see how to interpret the multiplication of complex …

Vectors and the Complex Numbers

Posted on Wed 14 March 2018 in Thesis • Tagged with thesis, quaternions, complex numbers

In the previous post, we first encountered the quaternions, giving an essentially algebraic definition by defining a quaternion as a set of four real numbers and defining operations to add, subtract, multiply, and divide these quadruplets. But my thesis work is in geometry, not algebra, and so it still remains …

Introducing the Quaternions

Posted on Thu 01 March 2018 in Thesis • Tagged with thesis, quaternions

In our last post we laid the foundation necessary to understand the quaternions, a 4-dimensional number system with some interesting properties that are applicable in higher-dimensional geometry. We didn't actually define or discuss the quaternions in that post, but instead discussed the real and complex numbers. It might be helpful …

On the Way to the Quaternions: Real and Complex Numbers

Posted on Thu 22 February 2018 in Thesis • Tagged with thesis, quaternions, complex numbers

In the introductory post to this series of posts explaining my thesis, we first heard about "quaternion-Kähler manifolds," a type of geometric object with special curvature properties. These special geometric properties are related to the algebraic properties of the quaternions, a 4-dimensional number system that generalizes the somewhat more familiar …