Faster Verification Process of Properties in Higher Dimensional Algebraic Structures using C++
General description of the project
Quaternions were originally brought to life by a man named William Rowan Hamilton. Hamilton was trying to make complex numbers more applicable in a 3-dimensional space since complex numbers are made up of a real number and an imaginary number. Hamilton initially theorized that quaternions consist of 2 imaginary numbers and one real number. He thought that he was making no progress then a few years later he managed to create the formula of a quaternion which is denoted by i² = j² = k² = ijk = -1. This study investigates the properties of algebraic structures, particularly quaternions, through the lens of operations defined on sets. We focus on the complex and computational challenges associated with verifying these properties, especially moving into a higher dimension.
Technologies
To address the workload involved in checking different properties, we investigate existing software tools that are available for verifying algebraic structures. We will analyze existing programming languages involved in checking associativity and commutativity in quaternions. Additionally, we will explore the efficiency of the algorithms used and look into the computational complexity. Our investigation will also identify potential optimization including whether rewriting the code in a different programming language like C++ could enhance performance and execution speed. We are going to use the set of complex numbers and set of quaternions to test our program and once we know it works then we will use that to investigate higher dimensional structure including octonions. To check for associativity we have to make sure to go through 64 checks on both sides and they would have to be equal to each other. This rule will also follow along with octonions which would be a total of 512 checks and we would want to avoid doing those checks by hand. We also want to check the previous codes to see if we do need to make that many checks and how long it takes to process this operation. We recognize the importance of identity elements, particularly the multiplicative identity. For example, any real number multiplied by 1 will give you back that real number respectively. Furthermore, having an identity element will have the inverse element whereas one-half times two would give you one which would mean that they are multiplicative inverses of each other. This illustrates that if the function has an inverse, the order of operation does not affect the outcome and will always arrive at the same result.
Explain project results
We anticipate that our program will efficiently conduct checks on associativity first, being the most tedious, and then commutativity. The program’s capability to perform rapid checks will significantly reduce the manual effort required for verification. Our goal in our program is to input specific collections and define specific operations to automatically execute the necessary checks and return results indicating whether the properties hold true. This will save time and minimize the potential for human error in calculations. Moreover, our software will quickly check and identify which definitions work for quaternions. By effectively validating these properties, our program could facilitate deeper research into the implications of quaternion algebra in various fields, including computer graphics, virtual reality, robotics, and aerospace.
Why it should be considered best practice?
In our research, we identified and analyzed key properties of these algebraic structures, emphasizing the multiplicative nature of real numbers and the implications of operations on quaternions. As we explore higher dimensions, we observe that in the fourth dimension, we lose commutative property in multiplication, as we proceed toward the eight-dimensional algebraic structure we also lose associative property in multiplication. However, checking whether the properties hold for particular algebraic structures and their defined operation is not an easy task and is very tedious. To show whether the associative property holds for quaternions, it would need to be manually checked 192 times, and for octonions, it would need to be checked 1,536 times. Thus it is important to quickly check if any of these properties are met when investigating algebraic structures in higher dimensions. While similar programs may already exist, our goal is to create a faster program that minimizes the amount of time it takes to check each property.
Highlights of your proposed presentation
In conclusion, our project is going to help mathematicians and physicists who are studying higher dimensional algebraic structures, including octonions, by allowing them to input any collection and operation defintions. This will allow them to quickly check whether their conjectures are true or false in a short period of time, rather than doing everything by hand. Our goal is to develop a faster method for checking property identities and operations associated with quaternions, as a result contributing to a deeper understanding of these complex higher dimensional algebraic structures. By making this more of an efficient verification process, we hope to open new opportunities for research and exploration in higher dimensions and promote advancements that may have practical implications in technology, science, and mathematics.
The Evaluation Committee will evaluate submitted proposals based on the following criteria. Each area will be rated on a scale from 1 to 5 (1= non-satisfactory; 5 =outstanding), for a maximum of 45 points.