Dissertation: Integrating Symbolic Multiplication and Creative Frameworks to Explore the Malleability of Mathematical Solutions Involving 1.5, π, and the Clock
---
## Abstract
This dissertation unites symbolic multiplication, creative transformation frameworks, and flexible start-destination solution paths to explore the malleability of mathematical solutions. By visualizing numbers and operations, creatively transforming constants like π, and applying these ideas to real-world phenomena such as clock rotations and the Navier–Stokes equations, the work demonstrates how mathematical rigor and creativity can coexist. The document is formatted according to current academic standards for mathematics dissertations, ensuring clarity, consistency, and accessibility.
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## Table of Contents
1. Introduction
2. Symbolic Multiplication System
3. The Number 1.5, π, and the Full Circle
4. Creative Constants and Transformations
5. Order of Operations and the PEMDAS Experiment
6. Layered and Multi-Start Approaches
7. The Role of Start and Destination
8. Application to Navier–Stokes Equations
9. Fundamental Principles and Infinite Solution Spaces
10. Navier–Stokes Equations: Rainbow Representation
11. Formatting Best Practices for Mathematics Dissertations
12. Conclusion
13. References and Further Reading
---
## 1. Introduction
The concept of a full circle, represented by 360 degrees or $$2\pi$$ radians, is fundamental in mathematics, physics, and everyday life, such as reading time on a clock. This dissertation integrates symbolic multiplication with creative frameworks—flexible constants, transformations, and multi-start solution paths—to deepen understanding of abstract constants and operations, and applies them to real-world phenomena and complex equations, expanding the solution space while maintaining mathematical rigor.
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## 2. Symbolic Multiplication System
**Visual 1: Symbolic Number Representation and Multiplication**
```
| = 1
|| = 2
||| = 3
|||| = 4
|‾‾‾‾‾| = 5 (grouped for readability)
Multiplication example:
|‾‾‾‾‾| |||| means 5 × 4 = 20
(visualized as four groups of five vertical strokes)
```
---
## 3. The Number 1.5, π, and the Full Circle
- A full circle = 360° = $$2\pi$$ radians
- Half circle = 180° = $$\pi$$ radians
- Define symbolic radian unit $$u$$ such that:
$$
1.5 \times u = 2\pi \implies u = \frac{4\pi}{3} \approx 240^\circ
$$
- On a clock, hour hand moves 30° per hour, so in 1.5 hours:
$$
1.5 \times 30^\circ = 45^\circ
$$
---
## 4. Creative Constants and Transformations
**Visual 2: Transformation Map**
```
[ π = 3.14 ]
/ | \
swap invert mirror
/ | \
[b_i = 1.43] [1/π ≈ 0.318] [d_i = 43.1]
Arrows labeled by operation type:
swap (blue), invert (green), mirror (red)
Legend explains each operation.
```
---
## 5. Order of Operations and the PEMDAS Experiment
**Visual 3: PEMDAS Circular Flowchart**
- Nodes arranged clockwise: P → E → M → D → A → S
- Each permutation highlights a different starting node with distinct color.
- Arrows indicate order of operations per permutation.
**Visual 4: Table of Results**
| Start Point | Order Followed | Result for $$a$$ | Notes |
|-------------|-------------------------|---------------------------|--------------------------------|
| P | P → E → M → D → A → S | $$a = \frac{e}{98}$$ | Standard PEMDAS |
| E | E → M → D → A → S → P | $$a = \frac{e}{98}$$ | Same result as P start |
| M | M → D → A → S → P → E | $$a = \frac{e}{98}$$ | Consistent across starts |
| D | D → A → S → P → E → M | $$a = \frac{e}{98}$$ | |
| A | A → S → P → E → M → D | $$a = \frac{e}{98}$$ | |
| S | S → P → E → M → D → A | $$a = \frac{e}{98}$$ | |
*Footnote:* Result remains invariant due to associativity and commutativity.
---
## 6. Layered and Multi-Start Approaches
**Visual 5: Layered Paths Diagram**
- Colored threads labeled P, NP, Q represent independent start points.
- Intersections mark "Hybrid Solution Zones" where solution paths combine or influence each other.
- Arrows indicate direction of solution flow.
---
## 7. The Role of Start and Destination
**Visual 6: Destination Alignment Schematic**
- Overlapping circles labeled with start points (A, B, C) and destinations (A, B, C).
- Matching start and destination areas highlighted for unique solutions.
- Mismatched areas indicate multiple or infinite solutions.
- Notes explain implications of each region.
---
## 8. Application to Navier–Stokes Equations
**Visual 7: Transformation Table for Navier–Stokes Inspired Equation**
| Transformation | Equation Form | Visual Cue |
|----------------|----------------------------------------------------------------------------------------------|--------------------------|
| Standard | $$3.14 \frac{d^2(n!)}{dx^2} + 1.43 \frac{d(n!)}{dt} + 43.1 n! = 0$$ | Standard equation |
| Swap | $$1.43 \frac{d^2(n!)}{dx^2} + 43.1 \frac{d(n!)}{dt} + 3.14 n! = 0$$ | Arrows swapping terms |
| Invert | $$0.318 \frac{d^2(1/n!)}{dx^2} + 0.699 \frac{d(1/n!)}{dt} + 0.0232 (1/n!) = 0$$ | Flipped fractions |
| Mirror | $$3.41 \frac{d^2(n!)}{dx^2} + 1.34 \frac{d(n!)}{dt} + 4.13 n! = 0$$ | Mirrored numbers |
---
## 9. Fundamental Principles and Infinite Solution Spaces
Mathematics is built on axioms and logic ensuring consistency, but flexibility in start points, transformations, and destination alignments creates an infinite solution space without loss of rigor.
---
## 10. Navier–Stokes Equations: Rainbow Representation
$$
\frac{\partial \mathbf{u}}{\partial t} +
\textcolor{orange}{(\mathbf{u} \cdot \nabla) \mathbf{u}} =
\textcolor{yellow}{-\frac{1}{\rho} \nabla p} +
\textcolor{green}{\nu \nabla^2 \mathbf{u}} +
\textcolor{blue}{\mathbf{f}}
$$
$$
\textcolor{indigo}{\nabla \cdot \mathbf{u} = 0}
$$
**Explanation of Terms by Color:**
- Orange: Convective acceleration (nonlinear transport)
- Yellow: Pressure gradient force
- Green: Viscous diffusion (momentum diffusion)
- Blue: External body forces (e.g., gravity)
- Indigo: Incompressibility condition (divergence-free velocity)
---
## 11. Formatting Best Practices for Mathematics Dissertations
- Margins: Left margin 1.25–1.5 in.; right, top, and bottom margins 1 in.
- Font: Times New Roman, Arial, or Computer Modern, 11–12 pt for text; headings up to 14 pt.
- Spacing: Double-spaced main text; single-spaced references and figure/table captions.
- Pagination:
- Front matter: lower-case Roman numerals (i, ii, iii, ...); title page counts as i but is not numbered.
- Main body: Arabic numerals (1, 2, 3, ...), starting at the first page of Chapter 1.
- Figures and Tables: Centered, labeled below the figure/table, numbered consecutively by chapter.
- Equations: Centered, with equation numbers right-aligned.
- Table of Contents, List of Figures/Tables: Automatically generated with page numbers and dotted leaders.
- References: Consistent citation style, single-spaced with double spacing between entries.
- LaTeX: Strongly recommended for mathematics dissertations due to superior handling of equations and formatting.
---
## 12. Conclusion
By integrating symbolic multiplication with creative transformations and flexible frameworks, this dissertation bridges abstract mathematical constants like 1.5 and $$\pi$$ with practical phenomena such as clock rotations and complex equations like Navier–Stokes. The combined approach visualizes multiplication and fractional units, applies transformations to constants and equations, and embraces layered, multi-start solution paths with flexible start-destination alignments. This synergy expands the mathematical solution space infinitely while preserving foundational rigor, highlighting mathematics as both a precise science and a creative art.
---
## 13. References and Further Reading
- Euler Archive: [https://eulerarchive.maa.org/](https://eulerarchive.maa.org/)
- Terence Tao's Homepage: [https://terrytao.wordpress.com/](https://terrytao.wordpress.com/)
- Stanford Dissertation Format Requirements
- UNM Thesis and Dissertation Guidelines
- CSUSB Mathematics Thesis Format
- Yeshiva University Master Thesis Guidelines
- University of Houston NSM Formatting
- Columbia GSAS Dissertation Template