Summer of Math Exposition

Billboarding in 3D graphics is a technique that has evolved significantly from its early use in video games to modern applications. Originally used in games like Super Mario 64 to conserve computational resources, billboarding involves displaying a 2D image that always faces the camera, creating the illusion of a 3D object. This technique has two main types: viewpoint-oriented, which directly faces the camera, and screen-aligned, which remains parallel to the camera’s viewing plane. The mathematics behind billboarding involves calculating vectors, cross products, and rotation matrices to ensure the billboard maintains its orientation relative to the camera. As scenes grew more complex, performance optimization techniques like culling and occlusion culling were developed to render only visible billboards, reducing computational load. Despite its effectiveness, traditional billboarding can struggle with complex scenes or objects viewed from multiple angles. Modern approaches, such as Neural Radiance Fields (NeRFs) and implicit neural representations, address these limitations by using neural networks to reconstruct 3D scenes from multiple images, allowing for more realistic renderings and interpolation between captured images. This evolution from simple 2D sprites to advanced neural network techniques highlights the significant progress in graphics rendering and the ongoing innovation in creating immersive 3D environments.

The problem specifically forms a bridge between quantum mechanics and graph theory. This video talks about how a problem inspired from quantum mechanics can be dealt as a graph coloring problem and interestingly, it's open.

Reviews divisibility rules for 2-11 and goes over the derivation

This video explains a heat-equation-based model for steak cooking optimization and uses an experiment to test its effectiveness. The model is based on Prof. Jean-Luc Thiffeault’s article “The Mathematics of Burger Flipping”: https://arxiv.org/abs/2206.13900 Timeline of the video: 00:00 General Introduction 01:49 Modeling the Steak 03:16 Deriving the Heat Equation 07:43 Boundary Condition 11:06 Nondimensionalization 13:05 Homogenize the Equation 15:31 Solving the Heat Equation 19:54 Modeling the Flip 25:51 Experiment Setup 25:35 Experiment of Optimized Steak Cooking 26:45 Result and Error Analysis 28:04 Conclusion

i couldnt rly understand isomorphism of objects before i visualized it with sets. here's a how

I explore and the idea of what numbers are and try and create a definition that includes numbers but excludes non numbers.

This video explores how to graph the Mandelbrot set, first by finding it's real portion, then complex, then even hypercomplex (partially). Some very interesting shapes are created.

The Complex Numbers are pretty well known, but their infinitesimal cousin, the Dual Numbers, is much less popular. However, I thought it would be interesting to combine the two into a single structure, but soon found there are multiple ways to achieve this depending on the exact properties you would want. With the first definition, the i and epsilon components do not commute, but in the second definition they do. You decide which is more exciting to you!

In this video it is explained how the ultrashort laser pulses can be viewed as superposition of monochromatic (infinite) waves. The video was inspired by the Nobel Lecture in Physics by Donna Strickland (particularly minute 24 in the linked video): https://youtu.be/sI_e7c085LM?si=2xKL4s3DDpIBHG1H

Details about a new algorithm to factor numbers in polynomial time.

An introduction to the fast growing hierarchy.

My take on the probability of singular outcomes with positive density. I am aware there are several errors in what I said and wrote. The part at 11:53, I meant that f(x) to be x squared, so that integrating k times x squared from 1 to 4 and setting it equal to one would give us a pdf. At 12:58, when I said the denominator is 1, I meant for the k times x squared function, for the appropriate value of k. Also, when I'm talking about the probability of x being between 2 and 3 on the continuum, the f(x) dotted lines in that range were supposed to look red. It seems like increasing the brightness of the video made the dotted lines appear black, which was not my intent. This is my entry for the Summer of Math Exposition 4, but unfortunately I ran short on time. Although I feel that this can be understood without first year college courses in Calculus and Probability, having a background in said courses will make the video even easier to understand.

a presentation on the agent model, why an agent uses and improves upon it's model of the world and why models in general though being inaccurate still prove useful also my first time filming and editing a video

I'm focused on practical solutions to Bertrand’s Paradox. I demonstrate how you should translate a problem from something ambiguous, like English, into something precise, like geometry. I assume the viewer has at least seen Bertrand’s Paradox before. My engineering advice applies to anyone who deals with "word problems."

By setting a=-1 and n: (n-1) you get the gamma function in the family of functions the final equation generates.

In the very young sport of streetlifting, athletes are in direct contact with the weight plates, as opposed to the sport of powerlifting or Olympic weightlifting. That's why it is important to select the weight plates and arrange them in a certain manner. My aim is to explore what makes an arrangement of plates good or bad and to create an alternative to the greedy algorithm of powerlifting.