Research and Ideation

Week 11: Initial Iterations of Zeta by Valzorra

Throughout this blog, specifically ever since Project Proposal 4 was introduced, I have continuously mentioned Zeta, the protagonist of As It Lies, giving little snippets of information about who she is and what she is like. I have been thinking about her as a character, what her major personality traits are, how she would behave, and what her story is. Based on those explorations, I have also thought about how Zeta might dress, how she might walk, what her posture is like and so on. I would like to take this opportunity to document my rather long iterative process for Zeta, how she was created, and how her visual style and appearance were determined. The very first step was to solidify Zeta’s personality, because that is what would drive her visual appearance and what she might wear. After all, an individual chooses their clothing based on what they would like and what would be of most use to them. I should also note that I did keep some of the details of the world in mind, more precisely the existence of the Alteration Implants, which may leave a visual trace.

She is intelligent, focused, creative, innovative, lighthearted, an adventurous risk-taker, a total gamer, slightly over-confident, extremely determined, and would do anything for the win.

In order to come up with as complete an idea of Zeta as possible, I decided to employ some of the techniques we learned during our Character Jam in Year 2. The specific method I used was to ask myself dozens of questions about the character, their habits, and their intricacies. Examples of the questions I addressed are “Where did the character grow up?“, “What is their daily schedule?“, or “What are their ambitions?“. The entire list of questions and answers regarding Zeta is detailed in the Characters Section of the Games Project Proposal. As they are rather relevant to the finished designs, I thought it would be better to include them there rather than repeat myself in both the GPP and the Blog. Once that process was complete I had a much better idea of Zeta as a person. She is intelligent, focused, creative, innovative, lighthearted, an adventurous risk-taker, a total gamer, slightly over-confident, extremely determined, and would do anything for the win. She frequents exquisite casinos, which would normally require a dress code, she casually hangs out with her friends in the city, and commonly stays at home to game as well. These three most common locations Zeta can be found in dictate that she would have at least three separate outfits we could see her in, namely her formal casino attire, her everyday clothing, and her clothing at home.

Casual Clothing: Initial Iterations

When beginning to visualize this character I thought I would begin with her casual and everyday clothing, as that is the attire we are likely going to see her in most frequently for the duration of the game. After all, it would be absurd and inappropriate to solve complex environmental puzzles in a formal dress. Thinking about what Zeta would wear, I imagined she would enjoy something rather comfortable that she wouldn’t have to assess and consider for too long. She is not the type of woman that would stay in front of the wardrobe and consider what to wear for more than a couple of minutes. That is why my initial iterations of her outfit featured rather baggy shirts, with plain jeans that would all work well with any combination of clothing. The back of the shirt is slightly elongated, because that would make Zeta’s overall silhouette more distinct and the flowing fabric would make her movement more dynamic. To contrast her baggy shirt, Zeta would need to wear fitted jeans or leggings that accentuate the shape of her legs. These would be a rather plain color such as black, grey, or denim, so that she can wear them with nearly everything else in her closet. In terms of shoes, Zeta is most likely to wear a comfortable pair of sneakers, which would go perfectly well with the rest of her casual and plain attire. Below I have included a carousel of fashion inspiration for the first set of iterations in the character design process as well as a few notable examples of the hairstyle I was going for.

Having described her clothing, its now time to discuss Zeta’s accessories and other physical characteristics such as her hair and body type. Zeta is rather fit and healthy, with a naturally slim figure, which does not require a lot of maintenance. Her body would likely be highly styled in the finished version of the game, which is why I have not focused on it too much just yet. In these initial iterations, I was trying to create a visually distinct and memorable character, someone that would look and feel special. I made Zeta’s hair rather long to help achieve that goal, and styled it similarly to other notable female leads such as Queen Daenerys Targaryen from Game of Thrones, Aloy from Horizon Zero Dawn, and Lagertha from Vikings (displayed on the images above). In addition to that, I made Zeta’s Implant rather visible, with it leaving a series of marks and scarring all over her body. Finally, I added the three dice that she would continuously use in the game as three long necklaces around her neck, so that she could keep them safe and close to her. Following these basic rules and decisions, I made a few different concept pieces of Zeta, with mild changes to the outfit, such as drawing her with a hoodie or with shorts. In terms of poses, I was mostly looking for what seemed most exciting and dynamic, what would be striking and grab the eye. The results from my initial concept art and drawing session are displayed below. Most of the images below were drawn by hand with the exception of the last one, which tried to breathe life into Zeta by adding color.

Reflection on Initial Iterations

The first set of iterations for Zeta helped me understand what direction I needed to take my design and art in.

Upon reflecting on the concept pieces I had done until this point, I grew increasingly displeased with them. Although at the time I thought I had very good reasons for making the decisions that I did, I came to the realization that those images were not a good representation of Zeta’s personality. She way very well wear the same clothes as they are quite casual, however, Zeta would not wear her hair like that, pose like that, or even bother with accessories. In order to achieve the hairstyle showcased above, Zeta would need to dedicate quite a substantial amount of her time in front of the mirror in order to get the braids right, which is something she just wouldn’t do. Additionally, most of the poses, with the exception of the right image on the middle row above where she is leaning against a wall, are all very staged and unnatural. Zeta is completely natural in her behavior, she is the heart of the company, and is not feminine at all in her day to day outgoings. What’s more is that although the images above are good concept pieces in themselves, they do not really reflect any sort of style that would feature in the game. Overall, there were a lot of problems with the first set of iterations for Zeta. However, they were a very important step in the design process as they helped me understand what direction I needed to take my design and art in. What I needed to do now was to start the process again, and try to get to the core of who Zeta is through my designs.

Week 7: The Riemann Hypothesis by Valzorra

Overall, Week 7 was rather slow as it was Reading Week, which meant that most of the week was free from any workshops or lectures. I took this time to catch up on some of my project proposals, which have already been published, and to document bits of research I did that are quite significant to Project Proposal 4 in particular. I will take this moment to note that the research itself was done last week, however, I am only now getting about to documenting it. Hopefully, it will all make sense and click together once laid out. Additionally, I wanted to take the opportunity to summarise and reflect on what Andy said during our one to one. Now, without further ado, let’s get into the research and updates.

The Riemann Hypothesis

The Euler–Riemann zeta function plays a crucial role in modern analytical number theory and has a variety of applications spanning across fields such as Probability Theory (*wink wink*), Physics, and Statistics. It’s basically a function whose argument can be any complex number other than 1, and whose values are also complex. Euler first studied this function as a real variable and was able to work out its values at even positive integers. In fact, the first even positive value of the function provides a solution to the Basel Problem. Riemann then expanded on Euler’s analysis of the function and established a relationship between its zeros and the distribution of prime numbers. What’s more is that from the Euler-Riemann zeta function stem a variety of other number series such as the Dirichlet Series and L-functions.

The real part and the imaginary part of the Riemann Zeta Function at the critical line.

Now that I have provided a very basic overview of the Euler-Riemann Zeta Function, I can move on to briefly describing the famous Riemann Hypothesis. The Riemann Hypothesis proposes that the Euler-Riemann Zeta Function has all of its zeros at negative even integers, which are all trivial zeros, and complex numbers with real part equal to 1/2, which are the more exciting non-trivial zeros. As the Euler-Riemann Zeta Function is closely connected to the distribution of prime numbers, if this hypothesis is to be proven it would completely revolutionise the way we interpret modern number theory and pure mathematics. What’s important to note here is that if this unsolved problem is proven it could open a series of doors to new ways we can think about mathematics and apply them in the sciences and in invention. Solving this problem would change the way encryption and computer system security functions fundamentally, which may be a reason why some might not want a solution to be found.

The Riemann Hypothesis has a series of parallels to the world and events examined in Project Proposal 4. In that universe, the equivalent of the Riemann Hypothesis, is the problem of humans only being able to use one implant, which is determined genetically. Even though there are a total of seven Alteration Implants developed, no one has managed to crack how one individual can use multiple at the same time. That is the major unsolved mystery of the time and the narrative of the game would revolve around what it would be like if such an important problem were to be solved. Additionally, one reason the protagonist is called Zeta is to echo Riemann Zeta Function and how she essentially managed to solve her universe’s equivalent of that problem. More details on the specific narrative will follow soon, however, even thus far I thought it was a nice nod to this area of mathematics and the potential consequences it may have.

Riemann, 1859

Andy’s Feedback

The Friday of Week 7 was dedicated to one to one sessions with Andy, and I was quite excited to hear what he had to say. As I was most fond of that idea, I presented Project Proposal 4, the game involving dice and managing environments based on what rolls. I gave Andy a very brief overview of Zeta’s world, of her character, and of the main dice-controlling mechanic. I described the special abilities I thought would be appropriate and how one would be able to shift between them. When I was finished with my explanation, Andy seemed pleased with the idea and even mentioned he is not completely sure what to add on to it. He asked me what I was most concerned about at that stage and I mentioned that although I have figured out how the world would work, I have not yet designed an actual level. Andy did not seem incredibly concerned about this and reassured me by saying that it sounds like a game he would like to try. He also recommended a book called The Dice Man by Luke Rhinehart, which revolves around a man who makes most decisions in his life by the roll of a dice. Overall, I was rather pleased with the feedback as Andy was not really able to punch any holes through the idea, which hopefully means that it’s rather solid. I look forward to getting more feedback on it from Adam and my course mates.

Week 7: Project Proposal 4 by Valzorra

Background

The core mechanic, the environment, and the story would all be based on mathematics, making this game a passionate love letter to the field.

Coming from my research on Probability and Chance, as well as Matrices and Markov Chains, I sought out to come up with a game where chance and probability were essential to the main mechanics (for more information on the research itself, please refer to Week 1: A Welcome Return, Week 2: Calculus and Cancer, Week 3: Research on Mathematics, Week 3: The Matrix, Week 4: Markov Chains, Week 5: Mathematics Visualised). With this in mind, I instantly thought of dice, and how dice could potentially be used and controlled within a game environment. Some of the questions I was thinking of were how would different dice impact gameplay, what would be the statistical chances of landing on certain faces, etc. (for more on Fair Dice, please refer to Week 3: Research on Mathematics). Keeping this idea of chance, probability, and dice as central mechanics, and coming back to my goal of creating a fun and empowering experience, with a few cool notches to mathematics (for more on this revelation, please refer to Week 5: Just for Fun), I thought it would be really exciting to assign special abilities to each side of the dice. Additionally, using my research on perspective and concepts like Desargues’ Theorem, I thought it would be quite cool to base the environment on those geometric principles (for more on that, head to Week 4: Dimension). The puzzles and environment in this game could stem from geometric proofs of theorems. This would mean that the core mechanic, the environment, and the story which is loosely based on the Riemann-Zeta Hypothesis, would all be based on mathematics, thus making this game a humble love letter to the field. I have detailed the specifics of the project below.

Project Overview

The action takes place in the late twenty-first century in a world not too dissimilar to our own, where humanity has come up with one of its greatest inventions, the Alteration Implants. There are a multitude of Alteration Implants or Alters for short, all of which give their users special abilities, such as Telekinesis, Teleportation, and more. All of the Alteration Implants developed immensely benefit society and are used in the workforce constantly. For example, the transportation industry has never been more punctual since the introduction of the Teleportation Alters. Each individual is genetically susceptible to only one type of Alter, which means that one person can only use one ability based on their unique genetic code, much like Blood Types. Because of this biological predisposition, individuals can usually only work in professional fields related to their Alter. The entire society is structured upon this principle, and the great scientific mystery of this time is to try and figure out how to enable people to use more than one.

You play as Zeta, a bright and ambitious young woman who has managed to crack this mystery. Zeta is also a highly-skilled and compulsive gambler. She lives in this universe’s equivalent of Las Vegas and plays competitively at the largest tournaments in the city. However, as skilled as she is she could never defeat the top ten players in the city, which is why she was looking for a way to gain an edge. Through her advanced mathematics and engineering skills, she figured out how to infuse standard six-sided die with the Alteration Implants available in this universe. After this breakthrough, she could use the dice in tournaments and gain an advantage over her opponents. Having a Telekinetic Alteration Implant herself, Zeta can also change the way the dice rolls, having freedom to pick and choose which side it would land on. However, unfortunately one of the players eventually catches on to what she is doing, starting a thrilling chase and mystery for Zeta, that will test her wits and determination. More on the specific details of the narrative would follow in future developments and posts relating to this project.

In terms of gameplay, what makes this project exciting is the intricate use of the dice and the opportunity to control chance and incorporate that into problem-solving scenarios. Zeta will have the power to use a total of three die. Whenever the player wants to use their abilities from the dice, they will roll all three and receive a certain value for each. Each value on the die will correspond to an ability. Using her Telekinetic Powers, Zeta would be able to change the way one of the dice rolls. She may change only one dice in the beginning of the game as she would not be particularly skilled at using her Telekinesis. As the player progresses through the game, they would have further control and would be able to change two and maybe all three die. Once the player has decided which dice they would like to change the roll of, all abilities may be used in the order the player chooses. For example, if I rolled a 3, a 1, and a 2, then changed my 3 to a 6, I would end up with a 6, a 1, and a 2. At that stage, I would be able to pick the order in which to use these abilities take place and I would have to make my way through the level based on that selection. Zeta is a gambler, so her risk-taking nature is perfectly incorporated into the gameplay. Additionally, this encourages the player to make the most of a situation with the hand they were dealt, connecting to larger themes of free will and choice.

Concept Pieces

I began some work on concept pieces and mechanics for Zeta’s abilities last week during our Tech Workshop session (for more on that, have a look at Week 6: Exploring Ideas and Mechanics). I have chosen a series of abilities that could potentially make it to the game, predominantly based on what the needs of that society would be. The Alteration Implants would have been researched and developed in order to help mitigate and alleviate some common problems within society, or to make existing practices more efficient. That’s why I have strove to ensure that all abilities have a good reason to exist within this society, which is why people would have invested into their research and development. I already knew I wanted to have some form of Telekinesis in order to help Zeta move the dice. To come up with other appropriate abilities for that purpose, I did a bit of idea generation based on the question If you could have any superpower or special ability what would it be? After I had a list of abilities that answered that question, I then proceeded to filter those answers with the question, Which of these abilities could be commonly used and benefit society? After answering that, I was left with a list of seven abilities, specifically:

  1. Telekinetic - Service Industry, Nursing, Building, Implanting Alters, Military

  2. Electric - Engineering, Computer Science, Chemistry, Robotics, STEM

  3. Teleportation - Transportation, Law Enforcement, Military

  4. Defensive - Firefighters, Law Enforcement, Military, Mining, Dangerous Professions

  5. Light - Agriculture, Ecology, Show Business, Entertainment

  6. Sound - Navigation, Military, Show Business, Entertainment, Ultrasonic Engineering

  7. Stopping Time - Secret Services, Military, Law Enforcement, Security

Idea Generation for all of the abilities functioning within this society.

Idea Generation for all of the abilities functioning within this society.

After narrowing down what all of the abilities within this society would be, I made a few storyboard-like sketches to showcase these mechanics individually within the game. The illustrations below do not really show how they would work in combination with each other, but the main idea is that these would be used to navigate through the environment. Additionally, levels would be designed in such a way that a multitude of combinations of abilities could solve the problem, resulting in a series of different approaches to the same situation. I have also explored a few different versions of some of these abilities, as I have yet to fully make my mind up about how they would function within the game. In any case, this would be something to explore within my prototypes over the coming week.

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Thoughts and Reflection

So far, this is definitely my favourite idea as it seems the most fun, it connects the best to my research, and overall has the potential to achieve all of the goals I want it to. I have yet to decide whether this experience would take place in three-dimensional or two-dimensional space as both have their pros and cons. If I make this a two-dimensional game, it would incorporate a lot of the things I am interested in, such as perspective and what type of information gets lost within perspective. My only reservation about making it two-dimensional is that I am not fully certain how to make it an exciting puzzle-solving experience that does not entirely rely on traditional platforming techniques. Whereas if I were to make the project three-dimensional, I would be making the idea slightly more generic and it may lose some of its mathematical flare, however, it is practically guaranteed to work efficiently. Overall, I am quite confident I will be taking this idea forward, at this stage it is only a matter of time to try and figure out how to present it. Hopefully some feedback from the presentations next week will help me get that clarified.

Week 7: Project Proposal 3 by Valzorra

Background

During our lecture on Markov Chains, James mentioned that they are primarily used in the fields of Marketing, Software Engineering, and Computer Science and have not necessarily been explored to their full potential in video games. That’s why I would be interested in creating a game which heavily features Markov Chains and individualised AI in its algorithms. That way the player would have a unique experience with the game, as its AI will continuously change based on their actions. For example, enemies may record and observe how the player behaves in order to predict their movements and become better managing them, which would involve Machine Learning, an area that I find rather exciting. Similarly, if the player happens to consistently overpower challenges set out within the game, enemies may become more vigilant and careful around them. The project would be all about adaptability of the AI and crafting a personalised experience based on what the user does, not dissimilar thematically to Project Proposal 2. Although there could be potential ethical considerations that might prevent this from happening, implementing some of James’s work on emotive AI into such a project might result in an interesting and unpredictable experience.

Project Overview  

I have always been fascinated by Edgar Allan Poe’s The Raven, which I briefly analysed in Week 3: Research on Poetry, and I believe some of the poem’s themes would connect perfectly with the idea of an individualised, possibly emotive AI. It deals with universal themes like consciousness, handling emotion, overcoming challenges, grief, loss, whether or not the mind and senses can be trusted and many more. Those themes in combination with my desire to create individualised AI, made me come up with the idea of a horror experience loosely based on Poe’s work where the main antagonists are seemingly unpredictable NPCs, whose personalities and behaviour would be dictated by Markov Chains. Additionally, what would make this experience even more impactful would be to design it for Virtual Reality as the medium would increase player immersion, thus making the characters seem more real.

This exponentially chaotic experience is essentially personalised horror based on sophisticated AI and themes from Poe’s work.

The events of this experience would start in a modern-day hotel room, with no specific explanation of what is going on, however, the environmental would suggest that we have just checked in (open suitcases, clothes on the bed, and so on). As the player explores the empty hotel room, they will discover the room door to be open, leading to a long hallway. Down the corridor there would be an old and shaken man, pointing towards an open elevator at the end of the hallway. As that is the only clear direction to go in, the player would naturally enter the elevator. At this stage, the elevator doors would close and could lead the player across a journey through the different floors. Each floor would reveal a different part of the hotel which the player can explore, such as the reception area, a bar lounge, an entertainment centre, and more. What’s more is that each time the player enters a new floor, it will appear as though the overall style and time era the hotel is in has changed. This will add to the overall idea of increasing chaos, horror, and disorientation. This connects to the themes of consciousness and questioning one’s own sanity found in Poe’s work.

It is precisely within those strange environments that the player will encounter the two main entities in this experience, the lovers Lenore and Arcadia. Lenore and Arcadia are seemingly all powerful beings who have a ghost-like presence in every single room. They would be the agents of chaos within this world, turning the tides from sanity to insanity and madness. They are everywhere, the span across space, across time, and the player will know that they are more powerful than them. Lenore and Arcadia would both adapt to the environment based on what the player does. For example, if they run through the whole experience without stopping to look around, they may become more aggressive and grab the player’s attention. Whereas if the player is rather calm and takes the time to look around at all details within the space, Lenore and Arcadia may be more subtle, leaving smaller hints about their presence. This would all be based on the variety of player personality types for example the Four Bartle Types, namely Killers, Achievers, Explorers, and Socialisers and the Unified Model (for more information, please refer to this wonderful Gamasutra Article).

The Keirsey and Bartle Model for Player Personalities

The Keirsey and Bartle Model for Player Personalities

Chris Bateman's DGD1 Model, which may feature in parameters that would dictate AI reactions

Chris Bateman's DGD1 Model, which may feature in parameters that would dictate AI reactions

I have constructed a very loose narrative that will be primarily conveyed through the environmental design and occurring events. In short, Arcadia is a talented singer, dancer, and performer, who unfortunately was never given the opportunity to perform in front of an audience as various managers and organisers would continuously deny her access. That is why Lenore, who is Arcadia’s demon-like lover, has created this hotel that perpetually traps its visitors. That way every night there would be an ever-expanding audience for Arcadia, keeping her happy and fulfilled. In a way, it’s a rather romantic gesture. Simulating the cyclic nature of The Raven and the idea of no way to escape circumstances, at the end of this experience, it will be uncovered that the player has actually been the old and shaken man from the beginning, guiding an unaware guest. At this stage, the player will have a series of options to reflect upon. They will have to decide whether these supernatural and time-bending events actually occurred, whether they have been trapped in this hotel with no escape, whether they have been playing as a mentally ill man who’s consciousness and senses betray them, whether this has all been a dream, or whether this has been a constructed scenario in the imagination of the old man. Either way, this personalised horrific experience would make players question notions of sanity, of how much we can rely on the senses, of consciousness, all built upon sophisticated AI.

Concept Pieces

Without going into too much detail, I have taken the liberty of developing visuals and a storyboard for a few of the key scenes in order to better convey the feeling and setting of this experience. Below I have attached a few sketches of Lenore, the main villain within this experience. She is an elegant beautiful woman, who has incredible confidence in her own ability. She is entirely devoted to Arcadia and would go to extreme lengths to keep her happy.

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Below I have attached an environmental sketch of one of the areas within the hotel. I thought an impressive lounge/bar area would be rather exciting to look at and explore. It is based on a few night clubs and bar spaces that I found quite atmospheric and that could potentially feel both welcoming and frightening depending on the events that take place inside. I also quite enjoy the architectural choices, especially placing the bar itself in the middle of the room, with seating areas around the walls, which I have applied onto the storyboards. Additionally, this would be the main area Arcadia performs in, so it should be quite interesting to have a look around the space and try to interact with the guests that would be in a very similar position to the player. This would also provide dozens of opportunities to feed information to the AI and make them take adapt to player choices.

Below are the first few pages of storyboard based on a scene that would take place within this performance area. I have not finished the scene just yet as I am not still undecided on whether this project will be going forward or not. The text around the images should explain what is happening in the scene and the actions/movements taking place. Hopefully, this will convey a small part of what Lenore is like and how she may behave when interacting with the player.

Thoughts and Reflection

I am quite fond of this experience because it has the potential to be a thrilling VR adventure, built upon sophisticated AI and some of the world’s greatest literary work. Additionally, I quite like the idea of slow descend into chaos, up until the world of this hotel seems to be falling apart. All of this to conclude with the player questioning what has happened and whether they are actually trapped within the hotel or their own mind. However, although the foundations of the project are based entirely on mathematics and code, I do believe it veers away from the overall theme of Maths, Chance, and Probability. Therefore, I am not convinced I will be taking this project forward, but I would still like to hear what others have to say about it before I make my mind up. Until then, onward with ideation and prototyping.

Week 6: Project Proposal 2 by Valzorra

Background

While researching Probability and Chance, specifically how to visualise the concepts, I came across a series of works in the field of Generative Art and Algorithmic Art (for more information, refer to Week 5: Mathematics Visualised). Looking at those striking images inspired me to think about an experience where one is completely surrounded by Generative Art and where one’s interactions with that environment continuously change it. For the most part, Generative and Algorithmic Art does not involve any interactive elements, but rather is restricted to either a static image or a short film. Therefore, Generative and Algorithmic Art could be elevated to a new level of interaction through the medium of video games. Additionally, I believe that there is more to be done in the field of Abstract Art Games and the ideas they could convey through algorithms, shapes, and form, so this project would provide a great way to explore that idea.

Project Overview

The project would essentially be a VR walk-through experience that allows the player to interact with its environment. Based on those interactions, the environment would change in accordance pre-determined mathematical functions, many of which would be based on Calculus (for more on that, please refer to the my work during Tech Workshop). As we have learned from Markov Chains and Matrices, in order to make procedural change, we oftentimes need a starting state. As the player transitions from their starting state and interacts with the environment, the environment will get continuously more abstract, changing with every movement, every touch of the player, based on functions like Probability Distributions, the Reimann-Zeta Function, Sine and Cosine Functions, and more. That’s the basic overarching goal and premise of the project, now on to the specific locations the players will explore.

Once they enter the experience, players will be placed in a tunnel like structure. This tunnel will be shaped to echo what the inside of the spine may look like. Within this tunnel, players will be able to walk, jump over obstacles, touch its walls, go up and down the tunnel in any way they please. As they progress through the tunnel and explore it’s structure, the environment will change with each movement and touch. Each action will have a mapped function to it, and those functions would then be applied to the tunnel, changing its shape, its course, even potentially destroying the tunnel. The experience will slowly transition from a detailed environment to an increasingly abstract and low-poly world. Finally, the player will reach a large open space, almost like the centre of the skull, where they will be able to look at and reflect on the changes they have made within the environment. That’s why each experience through this tunnel would be unique to every player. Additionally, this connects to a series of universal themes and ideas that would make the project that much more thought-provoking. It ties into the idea that actions have consequences, that everyone’s life path is different and those differences must be appreciated. It displays how mathematics is the underlying construction of the universe through an artistic medium. It would also connect to the idea of reflection upon one’s actions and how focusing on the big picture rather than the small details can sometimes be more revealing than anything else. Another exciting point about this project is that I believe it could work terrifically in a Virtual Reality Environment.

Concept Pieces

This would be an incredibly visual process, with no fixed narrative experience. Walking through the tunnel would generate a visual story and trace of your actions and movement. I have not created my own sketches of the concept as of yet, however, I have attached a series of images that will hopefully help convey how this may look visually. The first image below is a good example of how the space may look when the player first enters. I have not made any decisions in relation to colour just yet, however, that would be the overall starting shape of the tunnel.

Annihilation, 2018

Annihilation, 2018

As the player progresses through the tunnel, they will inevitably create patterns of trajectory, they may touch certain objects within, they might pick things up or destroy parts the environment. Based on those actions, the landscape of the tunnel and path may continue to change in ways such as the ones displayed below.

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As the player progresses through the environment further, its shape and outline will continue to change based on the predetermined set functions. It will inevitably get continuously more chaotic, more abstract and as previously mentioned, the world may even collapse on itself at that stage.

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At the end of the experience, the player will have a moment to reflect on all of the changes they have caused within this environment, which may end up looking something like the image below. This moment of reflection will allow the player to appreciate what their actions have caused in this world and whether or not they like what they see. Hopefully, it may also encourage people to reflect upon their own lives and consider whether or not they are happy with what they have so far. The question to follow would be if it’s too late to change or if things can be restructured.

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Week 6: Project Proposal 1 by Valzorra

Background

While I was researching Markov Chains and Matrices, I explored the concept of On-Page SEO, specifically what variables contribute to websites ranking higher in search engines. The conclusion of my analysis was that one can give numerical values to the key On-Page SEO Factors, and then estimate what changes need to be implemented to achieve the ranking one wants. In simpler terms, we can figure out what to change to rank higher based on where our site is at and where we want it to be (for the full analysis, please refer to Week 4: Probability Manipulation in SEO). However, although I have explored the relationships between the variables, their relative weights, and how they could fit into a Markov Chain, this process can get quite technical. Many website owners and publishers are not necessarily familiar with Matrices and Markov Chains, so the knowledge of how to improve their performance would require unnecessary training in a field that’s irrelevant to their work. However, I believe I may have found an answer to this problem through a new methodology for Data Visualisation (which is closely related to my research on the topic, found in Week 5: Mathematics Visualised). What’s more is that this method does not necessarily need to be limited to SEO, but it could rather be a universal visual training tool, which would mitigate the issue of working with raw numbers.

Project Overview

Most methods for Data Visualisation take place in two-dimensional space, such as histograms, point graphs, arc diagrams, and more. These techniques are quite efficient when working with a very limited number of variables and when the relationships between them are straightforward. However, it would be rather difficult to depict more complex and intertwining information. For example, On-Page SEO Factors roughly fit into three major categories, with a series of three to five subcategories, all of which affect each other differently. In order to represent this complex information, we need to take it into an additional dimension. By representing the data in three dimensions, we gain access to a series of solids, with a variety of faces, edges, angles, and vertices, all of which can be used to represent data. For On-Page SEO Factors, I have found that a tetrahedral shape represents the variables in the most clear and concise way possible (for more on tetrahedrons, please refer to Week 3: Research on Geometry). The three major categories of On-Page SEO would be represented by the base of the tetrahedron, while the pinnacle of of the solid would represent time.

Inputting different data points into this model would then change the tetrahedral shape into a different triangular polyhedron. A perfect tetrahedron would represent the ideal state of a website in terms of search engine success, while any other polyhedron that differs from the Platonic Solid would show what areas need to be improved. By inputting data from different websites into this model, we would essentially create a library of polyhedral shapes. That way if a website owner wants to improve their ranking in search engines, they wouldn’t need to learn a thing about Matrices or Markov Chains. All they would need to do is ensure that the shape of their website either matches the shape of a website they are aspiring to be like or is as close to a perfect tetrahedron as possible. What’s more is that this could also be a fantastic tool for education and training. For example, it could be presented to Marketing students, who would be tasked with creating a successful marketing strategy based on certain variables. The strategies the students have come up with, would then be inputted into the tool and all of the generated shapes would then be compared to each other and to the ideal. This would provide a clear and visual method for training and education, and could be applied to any field that uses some form of variables.

What’s even more exciting is the possibility to make this project a real-time interactive tool through the use of VR. Data Visualisation in VR has not been explored greatly and most efforts thus far have not taken great advantage of the medium. However, I believe this tool would be perfect for VR, because of its immersion and incredible interactive potential. Instead of inputting the data points for this model through a keyboard, one would be able to physically move them and change the data in the Virtual Reality Environment. As one grabs a data point and moves it about, changing its values, the entire shape would change then and there. Placing the model in VR would make the process a lot smoother and faster because moving the data points physically is extremely intuitive and can be much easier than typing different values in. At the end of the experience, the tool would provide concrete numerical data of what needs to be done to achieve the desired shape, allowing users to directly implement the suggested changes.

Concept Pieces

The concept piece below roughly shows how the model may look in the 3D environment. All the different coloured points represent data that could be shifted about to form a different shape, and how the connections between them would change. The larger concept piece represents a top-down view of the model, while the smaller one to the side shows what one of the faces would look like to the side, with its respective data points. The second concept image explains the process of taking the graph into the third dimension and constructing the shapes off of that, thus forming an elaborate library of shapes.

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Thoughts and Reflection

Overall, I am very satisfied and excited about this model because of its universality, usefulness and applicability. This is a method for representing any sort of data and can be used as a visual training tool for a multitude of fields. Additionally, it is a great way for the communication of complex data and its intertwining relationships. The tool can be developed and used on computers, tablets, phones, and it could even be a potential breakthrough when it comes to Data Visualisation in VR. The only problem with it so far is that this is very much focused on data and is not really related to games, which can make the model a bit dry for an FMP. Nonetheless, I do believe this is a project worth further research and pursing because of its incredible educational potential, and its value as a piece of clear and concise piece of design.

Week 6: Exploring Ideas and Mechanics by Valzorra

Idea Generation

We have officially entered the Phase 3, the first phase focused on actually building our ideas and creating a variety of prototypes for them. Before, I could start crafting anything, I needed to come up with a sufficient number of potential projects to work on. I already had a few ideas floating around, but this was the perfect time to actually put all of them onto paper and hopefully to come up with a few additional good ones. To do this, I decided to use the Lotus Diagram for Idea Generation, as it maintains ideas rather focused around a broader central concept and the method is also very efficient at forcing me to come up with about 70 concepts. I have attached the Lotus Diagram I came up with below, with the words Chance, Probability, and Maths in the centre.

The diagram itself looks rather intimidating due to the large number of ideas and writing on it, however, after examining everything I had come up with, I decided to focus on a few points that really inspired me. The major idea categories I chose from the centre were to use dice to determine the outcome of a situation and to then change those results, to find a better way for visualising complex data, to use the Reimann-Zeta Function as inspiration for a story, to generate an interactive world through mathematical formulas, and to create advanced and sophisticated AI players could interact with. Below I have attached a series of images featuring the ideas I liked and felt inspired to work on, which have all been highlighted in pink.

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After this process of idea generation, I felt quite happy with some of the projects that came to mind and was eager to start working on them as soon as possible. Many of the ideas can be combined, and turned into a more complex and exciting project, which is what I have attempted to do. I will detail each of these in the next couple of weeks in a series of project proposals that will explain my thought process, the connection to my research, and the project itself in a lot more detail. For now, the four major ideas that I would like to dive into are:

  1. A new method for three-dimensional Data Visualisation: I believe I may have figured out a way to represent complex data through the use of three-dimensional solids. This could be used as both an educational and diagnostic tool.

  2. A game environment generated by the players as they move and interact with it it. This would be an exponentially more chaotic experience that would take player actions, feed them into mathematical formulas and functions, and change the environment as they move along. By the end, each player would have created a unique world.

  3. An experience featuring sophisticated AI and based on Edgar Allan Poe’s The Raven. This experience would have predictive and potentially emotive NPCs that would convey a sense of fear in the player, which can be fun and thrilling in itself.

  4. In a world where each individual has one of a set number of special abilities, you play as a character who has managed to figure out how to use all seven of them. You were naturally born with Telekinesis and have infused a standard six-sided dice with the rest. This allows you to control how the dice rolls, giving you power over chance. Additionally, this could incorporate a story loosely based on what may happen if a solution to the Reimann-Zeta function was discovered.

Tech Workshop

We began the Tuesday of Week 6 with an exciting exploration of how Calculus and a bit of Trigonometry can be used in video games to create smooth movement and over the shoulder cameras. I found James’s explanation really straightforward and easy to understand and for the most part I didn’t really have any issues following along. I quite enjoyed this exploration because it showed a very handy practical application of the things we have been looking at over the past few weeks. Additionally, this algorithm and the logic behind it would work equally well in any game engine, because it’s based on mathematics rather than on in-built settings. What’s more is that this is a very elegant mathematical methodology for handling this issue in games design, and I look forward to applying it in some of my future projects. I would be greatly interested in looking and other such applications of Calculus within video games, so hopefully we will be examining more of the sort in the coming weeks. I have attached the notes from that explanation below.

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After the lecture on a practical application for Calculus within video games, we entered our Building the World session. James took the time to have one to one sessions with each separate team and to try and help us with our design process. I was one of the first to have a chat with him and as I was interested in developing fun mechanics players can mess around with, James suggested I create a few pieces similar to storyboards, which would describe how those mechanics would work. I decided to start working on a fun idea I had early in this week, which involved a variety of special abilities the player would be able to take advantage of. The complete idea will be described and documented over the course of the next couple of weeks. I also took this opportunity to expand upon the mechanics I was developing last week and to make the illustrations a little nicer and more clear. Additionally, my work from last week (Week 5: Just for Fun) featured a few different abilities in combination with each other, whereas here, I have showcased each one separately.

An environmental sketch to showcase the mechanics onto.

An environmental sketch to showcase the mechanics onto.

The slideshow below described a teleportation mechanics, whereby the player can choose a position within a certain range and be instantly teleported to that location. Another option that may make the mechanic a bit simpler would be to simply teleport the player a set distance in the direction the camera is looking at. This would make the teleportation mechanic less strategic, however, it may simplify things a bit if there is ever a risk of making the game too complex. Additionally, it would be interesting to consider what would happen if a player was to teleport directly on top of an enemy. So far, I believe they will simply not be given the option to do so, but it may be fun if an enemy instantaneously combusts when they are teleported on top of. All food for thought.

A defensive shield would function almost entirely as expected. The player would me a lot less susceptible to damage and enemy projectiles would not necessarily be able to collide with them. I envision the shield as a armour-like material, almost like shards, which would wrap itself around the player and protect them from any danger. The shield would have a relatively long duration and it would last for a certain amount of time. Another option would be to make the shield last onto the player until it is broken by enemies, however, making it based on a timer may be a bit easier to implement and manage.

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The last ability I was able work on during our Building the World session was a blinding beam of light that would essentially stun enemies. Initially, I thought that this beam could essentially rise above the ground and shine brightly, blinding enemies with it’s light. The player would be able to pick a target location to place the beam over that location. All enemies within it’s range would be unable to see for a set duration, however, that does not necessarily mean they would not be able to attack. This ability could be a double edged sword as on one side it stuns enemies, but on the other, it definitely draws attention to the player. I may make another version of this in the future as I am not incredibly pleased with it just yet, but nonetheless, so far so good.

Week 5: Game Research Document v0.1 by Valzorra

Update: I have finally come to my senses and realised that what I really want to do is to to simply create a fun and entertaining game, which has some exciting notches to mathematics, without taking itself too seriously. I want people to enjoy this experience, to have a fun time, to go on an adventure through its gameplay, to immerse themselves in its world. This project will be a subtle love letter to mathematics, incorporating its principles into its mechanics and structure, while also letting people have fun. With this new mindset, I look forward to going into Phase 3, and exploring dozens of ideas on how to achieve all of this. Now, moving on to the Game Research Document.

The Practice-based Research Game Project Document is meant to be an evolving document of research, thoughts, and design decisions on our game idea, and it will be developed throughout the course of the semester. This is my very first version of the document, and I was only able to actively fill out three main sections of it, the Abstract, the Goals, and the Target Audience. As we progress through the year, the documentation will hopefully continue to be filled and flushed out in more detail. However, for now this is a very brief summary of the key points of my research so far and why I find certain aspects of it rather exciting. Without further ado, enter v.0.1.

Abstract

The study of Chance and Probability is essentially a way of looking into the future through mathematics, which can be incredibly powerful.

Mathematics is an incredibly diverse and exciting topic with an infinity of intricacies, principles, patterns, proofs, all with a watertight underlying logic. It is the essential basis of all other sciences and it finds its way even in fields such as art and design. The abundance of information in the subject meant that I needed to choose a few key focus points within the field to explore further, which was rather difficult as there are numerous topics of interest to me. I am absolutely fascinated by the idea of Chance, Probability, and manipulating seemingly random events through certain rules and principles. That’s why I greatly appreciated looking into things like the Binomial Distribution Formula, Matrices, Markov Chains, and Fundamental Calculus in our Tech Workshop sessions, which I then went on to research and examine further (many thanks to James Stallwood for bearing with me). All of these different methods directly relate to how one can manipulate probability and how to make accurate predictions about the likelihood of certain events from occurring. I find this absolutely incredible, as this means that one can utilise the results from the methodologies listed above to better their decision-making process both in a game and in real life. It’s essentially a way of looking into the most likely future through mathematics. Not only can we peak into the possible future, but we can also alter it through the way we structure data. Maths gives us the power to make decisions based on the results we want, we can transition from the current state of affairs to a desired one with certainty. For example, with Matrices and Markov Chains, one can calculate exactly how much to invest in a marketing campaign in order to get the most optimal boost of clients. This means that Mathematics can help a business flourish through a few simple equations. The same logic can be applied within an interactive environment, it can give users control and hindsight into any seemingly random situation. It’s all about representing data, controlling data, seeing where we are at, where we want to be, and how we shift those probabilities in such a way that we obtain the desired outcome.

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In the discussion of Markov Chains, and Matrices, I have explored the logical methods by which one can control chance and probability. However, these methodologies require a fairly sophisticated level of understanding of both Mathematics and Functions with Matrices. Based on that, I wondered how one would actually visualise complex data structures in a clear and concise manner. Answering this question lead me to the wonderful world of Data Visualisation, which is an entire field dedicated to these problems. Data Visualisation is about clear communication and ensuring the untrained eye can easily tell what the information is and what the intertwining relationships between its variables are. This is an essential link to the study of Chance and Probability, because once those can be visualised, then they can be altered and manipulated by anyone, even those unfamiliar with the specific mathematical rules. Additionally, Data Visualisation involves thinking about problems from different point of view and perspective, which is entirely what mathematicians do when solving problems. It’s about presenting solutions to problems by looking at them in a different light, about showing those solutions in a comprehensible manner, and about educating on the issues presented. Although the field is quite well-researched, I do believe that more can be done, specifically in three-dimensional Data Visualisation and Data Visualisation in VR, which is largely unexplored.

Data Visualisation: Frequency of Text Messages Within a City

Data Visualisation: Frequency of Text Messages Within a City

Finally, what makes Mathematics incredibly exciting is the idea that the same problem can be looked at from a variety of equally valid fields and perspectives, which could all work together to bring up a solution. Geometric mathematicians may focus on the volume of a coffee mug, those interested in algebra might consider how all of its dimensions relate to each other, the Calculus bunch might want to calculate the instantaneous rate of change on the curve of the handle, while topologists might loudly explain that “This is the same as a donut!“. All of these different professionals offer a very different angle on the same problem, and are all perfectly sound in their logic. I find this diversity, this variety in how you can go about the solution of a problem truly inspirational, and I would love to incorporate that sort of thinking into my design practise. Specifically, this sort of thinking could be well-incorporated into level design and mechanics, so there is the potential for my FMP to turn into a purely mechanical experience. Overall, I am very excited to begin work on a specific project and to carry on with ideation and actually solidifying a game.

Goals

As I have not quite made up my mind on what project to go forward with, I have listed a series of goals I would be happy to achieve this year.

  1. To create a fun game, simply for the sake of fun and entertainment. I would love to create something that people can come home to and enjoy, relax to, and maybe even feel powerful in. I want to create a piece that will that will get people happy and excited, that will form friendly and competitive discussions.

  2. To demonstrate that chance and probability can be manipulated through the use of mathematical formulas and principles. Individuals can predict seemingly random events and incorporate that knowledge into their decision-making process within a game.

  3. To come up with a better way of visualising data and manipulating variables in the third dimension in order to help people better understand information and and its incredible predictive power.

  4. To challenge modern notions of level design and to provide an experience with a series of possible solutions based on the idea of controlling chance events within an environment. To challenge people to think critically about problem-solving and to analyse the level and environment in order to solve it. Hopefully, this would lead to an incredibly satisfying experience.

  5. To improve my own skills in the fields of Level Design, 3D Modelling, Programming and to efficiently incorporate Mathematics into both the mechanics and narrative elements of a game.

Visualising GPS Data From Various Marine Vessels.

Visualising GPS Data From Various Marine Vessels.

User Experience: Audience

Although it is a bit early to say for certain, I would like the Target Audience of the project to be quite wide, possibly Teens and upwards. As I previously mentioned, I want to create a genuinely fun experience with a series of exciting levels that involve or are based around a variety of mathematical principles. The reason I am setting a minimum age of teen/pre-teen is because I do not believe younger children will necessarily be able to appreciate the mathematical logic and principles involved in certain aspects of the game. Additionally, creating a game suitable for everyone means that I would have to restrict myself in some capacity as to the content of that game, which I would not like to do at this stage. If the game happens to involve violence to a certain degree, I would like to have that option available and not tie myself to a universal ranking. However, coming back to the beginning of this paragraph, I believe it is far too early to tell for certain who this project would be most suitable for. All that can be said at this stage is that I would love it to be open to as wide an audience as possible, people from both genders, all sexuality, ethnicity, geographical location, etc. The beauty of mathematics is that it is universal in its logic and it can be appreciated by anyone around the world with basic knowledge of it.

Week 5: Mathematics Visualised by Valzorra

Visualising Probability

We’ve already discussed Chance and Probability in great detail in Week 1: A Welcome Return, Week 2: Calculus and Cancer, Week 3: Research on Mathematics, Week 3: The Matrix, Week 4: Markov Chains, and Week 4: Probability Manipulation in SEO. Although all of that study has given great understanding of how one can control and manipulate probability to make educated and diverse design choices, I haven’t really discussed how Probability is visualised in Mathematics (aside from the Probability Trees discussed with James). In order to visualise probability mathematicians and algorithmic artists turn to Probability Distributions.

The foundation of all Probability Distribution is the Uniform Distribution. In Uniform Distribution there is an equal chance of achieving each result from a certain set or range. For example, with Fair Dice (for more information go to Week 3: Research on Mathematics) we would use Uniform Distribution to visualise the chance of landing on each number of the dice as by definition the chances are equal. Uniform Distribution is also the default for nearly every random() function in programming. It’s interesting to note that when although Uniform Distribution attempts to follow true randomness, it does then to form small clusters and clumps in certain areas. Nonetheless, this Probability Distribution is fantastic for achieving an equal coverage of an area and would be visualised in one dimensional and two dimensional space as illustrated below.

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Gaussian or Normal Distribution is extremely popular both in nature and in everyday human activity. In Gaussian Distribution, commonly referred to as a Bell Curve, the data tends to be focused around the centre values with no bias towards the left or the right. In this Probability Distribution, values closer to the mean (which is also equal to the median and the mode) tend to be chosen more often. What’s noteworthy about Normal Distribution is the idea of Standard Deviation, which simply indicates how spread out the values are. Typically, 68% of all values fall within one Standard Deviation of the mean on either side, 95% of values fall within two Standard Deviations from the mean, while 99.7% fall within three. Visually, Normal Distribution tends to focus around the centre, with few values spread out on the sides as shown below.

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Another way of visualising probability is through the Power Law Distribution, the most popular of which is the Pareto Distribution. Power Law Distribution is exponentially skewed towards the minimum possible value. These type of Distributions are most commonly used to describe observable phenomena. The most notable example that uses this format is for the Distribution of Wealth in Society as that illustrates how a small portion of the population holds substantial amounts of wealth. Aesthetically, this distribution achieves a good balance between different sized objects. Pareto Distribution in both one dimensional and two dimensional space is shown below.

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Probability in Generative Art

The Probability Distributions mentioned above are significant to the way we represent and structure data and to the study of Data Visualisation. However, there are also a number of artworks that take advantage of their properties and logic, specifically Algorithmic Art and Generative Art. Both of those art fields take advantage of autonomous systems such as machines, computers, language rules, genetic sequences, algorithms, and more. Due to the nature of the autonomous systems, the artworks are often created with an element of chance and probability, which can be controlled by the artist or designer to achieve certain effects. The idea of controlling chance and probability through algorithms and mathematics strongly ties into what interests me about the field, which is why Generative Art is perfect visualisation of that concept. Below I have provided a few notable examples of Generative and Algorithmic Art by Tyler Hobbs, which illustrate the field’s relationship to order and randomness, to aesthetics and mathematics. The works below take advantage of Gaussian Distribution and Power Law Distribution in order to obtain a good balance of shapes and density within the works, making them appear clean, precise, yet packed with information.

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Data Visualisation

Data Visualisation is the study of the representation of data in the most clear and easy to communicate manner possible. Oftentimes it involves taking vast amounts of data, involving complex and intertwining relationships, and representing it in a visual and digestible format. While Matrices and Probability allow us to manipulate data, Data Visualisation allows the untrained eye to work with that information and understand the contents of a Matrix, for example. There are dozens of different and innovative ways to represent data, both in two-dimensional and three-dimensional space, including Arc Diagrams, Density Plots, Chord Diagrams and many more (for more examples, have a look at Week 2: The Four Themes). Data Visualisation incorporates elements of Graphic Design, User Experience Design, User Interface Design, Mathematics, the Sciences, Art and is all round an incredibly useful discipline for visual communication. Below I have attached a few notable examples and creative approaches to Data Visualisation, which could also be referred to as Data Art. What I find incredibly exciting about Data Visualisation is the idea of clear and concise communication of complex ideas through the power of good design. This is a method of helping people understand information, it can be incredibly educational, and can help train people who have no prior knowledge of the subject matter. Therefore, Data Visualisation is a field worth investigation for both designers and educators.

In the project below, Datamatrics, Ryoji Ikeda has taken enormous volumes of data from computer generated code from software and hardware errors and bugs. All of these errors have then been represented in a variety of two-dimensional and three-dimensional formats. The project is not a traditional use of Data Visualisation as it is actually aimed at representing what can often be the chaotic nature of code and computer software. There has been a later iteration of this specific project, which explored the concept in purely binary values and further reiterated the idea of simplicity and chaos within the realms of technology.

The work below by Marcin Ignac represents the Global Sea Transportation Network is entirely based on the paths of sea transportation vessels and their actual GPS Coordinates as they travel around the world. This project is especially connected to Matrices as the data from those GPS Coordinates would be recorded within a Matrix. Therefore, this is a terrific example of how Matrices can be presented in a very easy to understand and digestible format, allowing for more clear communication of the data. What’s more is that by adding new routes and new matrices within this system, it would be incredibly easy to observe changes and estimate where certain ships are meant to be at any given time.

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Platonic Solids Transformations

I have gone into detail over the Platonic Solids and why they are exciting in some of my previous posts (Week 3: Research on Geometry, Week 3: Research on Mathematics). However, what’s even more interesting is that they can actually be used as the basis for a near infinite number of other complicated shapes through the use of Geometric Transformations. I found a fantastic tool online created by Marcin Ignac, which can edit Platonic Solids to create a multitude of more complex shapes. Some of the techniques used in the tool include splitting the faces of the Platonic Solids into triangles, changing the heights of those triangles. Additionally, shapes can be merged together, their faces and geometric properties working in combination with each other to create a new solid. Overall, the tool is an absolutely fantastic demonstration of Transformations within three-dimensional space and what’s more is that any figure can also be exported as an OBJ. There transformations can be turned into separate mechanics and functions within a game, making the tool and the mathematics behind it a method for idea generation. I highly recommend exploring the Platonic Solids Editor as it provides valuable insight into many of the geometric properties discussed throughout the blog.

Week 4: Dimension by Valzorra

Representation of the Third Dimension Into the Second Dimension

The representation of three dimensional space on a two dimensional surface has been explored over hundreds of years in a variety of artistic attempts. Mathematically, the idea of perspective and vanishing points is what is exciting about the representation of these dimensions. The basic idea of perspective is that objects appear to be smaller the further they are away from the viewer, and there is an accurate mathematical way to represent perspective. The most basic way to do that is to use a single vanishing point, which used to traditionally positioned in the centre of the canvas. With this technique horizontal lines are perpendicular to the canvas, while all vertical lines lead up to the same point in the centre. To make the perspective more exciting, artists later included multiple vanishing points, sometimes positioning them outside of the canvas. However, with more than one vanishing point, the horizontal lines can no longer be perpendicular to the canvas as they would not appear horizontal in the world of the paining or image. When there is more than one vanishing point, we refer to the line connecting those points as the vanishing line.

There are other ways to give the illusion of perspective other than using traditional vanishing points. Introducing Desargues’ Theorem, which relates to how triangles can be in perspective of each other without the use of vanishing points. In order to explain the two main notions of the theorem, let’s introduce two triangles ABC and abc. Desargues’ Theorem states that if the points Aa, Bb, and Cc all converge to the same point, then the two triangles are in perspective from a point. Now, to explain the other notion of the theorem, let’s call the meeting point of AB and ab = D, the meeting point of BC and bc = E, and the meeting point of AC and ac = F. If D, E, and F all fall on the same line, then the triangles are considered to be in perspective from a line. Desargues’ Theorem is a keystone notion in the field of projective geometry , which deals with the representation and transformation of geometric objects. If you would like to explore the proof of Desargues’ Theorem, please refer to the video below as it explains the concept better than I could ever hope to.

One of the most notable artists who have tackled the idea of representing the change from second to third dimension is M.C. Escher, who’s work is absolutely fascinating mathematically. Specifically, his experiments with Tessellations (which are simply infinitely repeating mathematical patterns, usually closely fitted together) continuously merge and play with the idea of dimensions and the constant shift between them. In the example below, Escher represents the cyclic nature of life by depicting the perpetual existence of the crocodile, which manages to escape its position, reach unknown heights, only to swiftly return and crawl back into position, repeating the whole process once more. The reptile manages to become a higher form of existence by entering another dimension, however, that is incredibly short lived. Additionally, M.C. has placed another object in this Graphic which shares a similar transitional nature, and that would be the dodecahedron, one of the regular polyhedrons. As discussed in a previous post, this is one of the few objects in existence that retains its mathematical properties while shifting between dimensions, thus further reiterating the main idea behind the reptiles.

Reptiles, 1943, M.C. Escher

Reptiles, 1943, M.C. Escher

Representation of Higher Dimensions

Representation of the fourth dimension can be excessively challenging, however, there are a few notable ways of visualising it, specifically through the Hypercube. One of the most famous examples of a depiction of the Hypercube is Salvador Dali’s Corpus Hypercubus, where he used a net of eight cubic cells glued to each other. Another possible method, commonly known as projection, features one cube located in the centre of another, with their corners joined together by edges. However, all of these are mere representations of the Hypercube through one format or another, while true vision of the fourth dimension has yet to be achieved, if at all possible. What’s interesting to me about this specific section of geometry is how these different shapes interact with each other, what their core principles are, how they help shape our understanding of dimensions and the main pillars of their construction. The transfer of information through dimensions and the existence of extraordinary shapes we cannot even imagine fascinates me and motivates me to look into them even further.

Crucifixion, 1954, Salvador Dali

Crucifixion, 1954, Salvador Dali

Mathematically speaking, the fourth dimension (and other higher dimensions) could be represented through the use of matrices, filled with Cartesian Coordinates and data points. The four vertices of a square can be represented by (0, 0) (0, 1) (1, 0) (1, 1). Adding one dimension, we can represent the cube as (0, 0, 0) (0, 0, 1) (0, 1, 0) (1, 0, 0) (0, 1, 1) (1, 0, 1) (0, 0, 0) (1, 1, 0) and (1, 1, 1). Adding a dimension one more time, and we would have the mathematical representation of the fourth dimension given by (0, 0, 0, 1) (0, 0, 1, 0) (0, 1, 0, 0) (1, 0, 0, 0) (0, 0, 1, 1) (0, 1, 0, 1) (0, 1, 1, 0) (1, 0, 0, 1) (1, 0, 1, 0) (1, 1, 0, 0) (1, 1, 1, 0) (1, 1, 1, 1), a total of 16 vertices. In order to represent even higher dimensions than the fourth dimension, we would simply need to add an additional coordinate. What’s even more exciting is that through the properties of matrices, one could then upscale or downscale the n-dimensional object, transforming it into any desired size.

Moving on from one dimension to another results in the loss and gain of certain information about those objects.

However, working purely in numbers is not very visual and does not provide a very intuitive idea of how to think or work with four or higher dimensional objects. I have previously explored the geometric properties of certain Polychora in Week 3: Research on Geometry, so feel free to explore that section of the blog for further visualisation. Looking into a way to make higher dimensions more intuitive, I came across this fantastic video that combines analytical and geometric methods of thinking about the fourth dimension, specifically a 4D Sphere. The basic method detailed in the video is to use a series of sliders in order to represent the points in four dimensional space, rather than to use strictly coordinates or strictly geometric shapes. Furthermore, what one will discover through this method is that in higher dimensions, the geometric shapes seem more counter-intuitive, which would force mathematicians and enthusiasts to be very creative when working and explaining their properties. More detail on the subject can be found in the video itself, but for my research purposes, the methodology of visualisation and representation is most significant.

Thoughts and Reflection

What I find really interesting about the idea of dimensions and going in between dimensions is this idea of information. Moving on from one dimension to another results in the loss and gain of certain information about those objects, which is absolutely fascinating to me. For example, a 2D representation of a 3D cube could not possibly display all of the edges as having the same size, as then the object would not appear to be a cube anymore, due to the rules of perspective. Additionally, the idea of different perspectives showing different sides of the same object, revealing new data about that object along the way could be perfectly translated into a game mechanic. What’s more is that this is the primary way in which one can create art with optical illusions and potentially use those in an environment. Overall, I am rather excited to see if this will go any further, but for now, onward with the research.