Tidbits

Kids born on March 14, 2015 are highly mathematical.

Tidbits

"Of course not all Asians are good at math. Why do you think some of us vote Democrat?"

Tidbits

Math in action: desert cup layout. Goal is to fit as many round desert cups on a tray as possible. I got to do this problem while working food service.

Previous person had made a 3x3 grid, 9 total, with the cups were next to each other. It was very inefficient. By staggering the cups, I was able to do 4 rows of 4, nearly double the capacity. According to the guy in charge I was the first one to think of that. That's both a personal ego boost and kind of depressing since it's so obvious.

Ordinal Numbers and Letters

In terms of spacing, incrementing by letters allows greater distinctions than numbers. Given a single space, digits can do ten distinctions: 0-9. Letters have 26: A-Z. Add a second space. Numbers give you 100, but letters can do 676.

Interesting that, by nature, we tend to use numbers over letters for sorting, despite the greater economy of a letter-based ordinal system.

Tidbits

Who's the best singer when it comes to different kinds of curves?

Harry Conics, Jr.

Tidbits

What's the best book of math poetry?

"Leaves of Gr-asymptotes."

Tidbits

When Bertrand Russell wanted to insult someone, he called them a Kant.

Tidbits

How did the math teacher encourage his students?

"Don't believe in yourself. Believe in e! Believe in the e that believes in you!"

Tidbits

When teaching logarithms:

Who the ln do you think I am?

Tidbits

What does the logarithm say about e's power level?

It's over NINE THOUSAAAAAAND!!!

Tidbits

What did the apathetic dependent variable say to the independent variable?

"I give no functions about you."

Tidbits

Idea for a math word problem involving inequalities: Goku's power level can be modeled by a function. In order to defend the Earth from Vegeta and Nappa, Goku needs his power level to be OVER NINE THOUSAAAAAAAAAAND. Calculate how long he has to train.

Ambiguity

1-2

1
Or
2

-1
Or
To

Tidbits

Joke told in Algebra 2 while covering the inverse of exponential functions:

How are logarithms and Skrillex similar?

They both like to drop the base.

Self-Referential Sentences

This sentence is not a sentence.

This sentence is illiterate.

This sentence has no words.

Do you care about this Senten and what it says?

This Senten says many interesting things.

This "Senten says" is false.

THIS SENTENCE WOULD NOT BE JUDGED TO HAVE HIGH COMMUNICATIVE VALUE BY THE AVERAGE ENGLISH SPEAKING PERSON WHO MIGHT HAPPEN UPON IT, BUT ATTEMPTS TO COMPENSATE FOR THAT CLEAR AND OBVIOUS SHORTCOMING USING THE ATTRIBUTE OF VOLUME.

Still a Better Love Story Than...

A new four book fiction series for teenage girls to get them hooked on math:

Book 1: Tangent
Book 2: New Min
Book 3: Ellipse
Book 4: Taking Domain

It features a love triangle involving a circle, trapezoid, and a rectangle (who sometimes turns into a parallelogram).

Tidbits

Did you hear about the math student who was allergic to infinity?

Yeah, they say his mouth was full of Cantor-sores.

Recursive Inigo Montoya's

Name a character: "Hello. My Name is Inigo Montoya. You killed my father. Prepare to die." It'd be funny since his name isn't Inigo Montoya. In fact, to make the sentence true, you'd have to replace "Inigo Montoya" with the phrase "Hello. My Name is Inigo Montoya. You killed my father. Prepare to die." This makes it:

"Hello. My Name is

"Hello. My Name is Inigo Montoya. You killed my father. Prepare to die."

You killed my father. Prepare to die."

However, the new phrase is still incorrect, so you have to insert it again:

"Hello. My Name is

"Hello. My Name is

"Hello. My Name is Inigo Montoya. You killed my father. Prepare to die."

You killed my father. Prepare to die."

You killed my father. Prepare to die."

But that still isn't right! Eventually it becomes a long series of "Hello. My name is" over and over again for infinity, followed by an infinitely long string of "You killed my father. Prepare to die." But you never get to hear that part.

Steampunk Math Geeks

Steampunk is growing in popularity right now. However, there are far too few steampunk fans who also play table-top RPG's. There are also too few steampunk fans who are simultaneously math geeks. This needs to be fixed. After all, the development of modern logical math is one of the key themes of steampunk.

Bertrand Russell, Georg Cantor, Alfred Whitehead, Henri Poincare, Gottlob Frege - these are all figures of history who bucked prior trends to blaze new trails in math. The philosophical debate of Poincare and Frege came to fist fights in bars, as mathematicians hotly contested the way to view not just numbers but the entire field itself. It was a period of intense mental energy and coupled with the sensation that, ultimately, everything was knowable through the power of math (and by extension, science).

It doesn't get more classically steampunk than that.

Game Systems 1: Games as a Formal System

It is said that discussing the familiar in unfamiliar terms opens up new avenues for thought and creativity. That is my goal here. By approaching games, especially RPG’s, the same way one would a theorem or scientific principle, my hope is to inspire others to rethink of their perception and conception of games themselves. It might prompt people to develop in ways they never thought of before and to explore new mechanics they discovered by codifying what just came intuitively for them.

I’ll start with something very general, but also very vital: formal systems. Formal systems have three major components (and I am borrowing heavily from Douglas Hofstadter for these terms):

1. Axioms
2. Typographic decision procedures
3. Theorems

Math is the most well known formal system. Geometry, for example, consists of Euclid’s famous axioms (with non-Euclidean Geometry altering one of them), a system for deriving proofs based on those axioms, and the theorems that follow from those axioms.

Games are also a formal system. We tend not to think about them as such, preferring to view them as crafts or works of art. And there is indeed a strong element of art to games, from the ideas present and the execution of their delivery. But in starkest terms, games operate on the same principles as math, language, and other systems, albeit at a less abstract level than fundamental elements. By realizing this, it can be possible to define and formalize a number of otherwise elusive terms we use in gaming, such as immersion and elegance.

In a game, the axioms are the materials required to play. The decision procedures are the rules of the game, outlining how the axioms can be utilized. The theorems are then the valid moves that can take place within the game. For example, in chess, the axioms are the board, the pieces, and the two players. The decision procedures are the way the pieces move, removal of pieces, win conditions, etc. The theorems are the set of valid moves a player can make on their turn, such as castling or getting out of check.

A trait of formal systems is the occurrence of isomorphism. Essentially, when the mind is confronted with a set of highly abstract concepts, such as the symbolic language of the 1900’s logicians, we will assign meaning to the patterns. Often these meanings will be drawn from personal experience and knowledge and used as a way to grant additional meaning to the patterns we see. For example, when school children are taught addition with the plus sign, they are prompted to think of the physical act of combining two separate piles of things into one pile. This creates an isomorphism: when you see the abstract symbol, +, think of two piles being combined.

This has the benefit of allowing us to understand something very abstract and foreign, by couching it in the familiar. However, the problem with isomorphism is that we will want to assume the system behaves like the thing we associate it with. This can cause issues in math, where not everything behaves as in the real world. For example, if you add the set of all the composite numbers and the set of all the prime numbers together, your isomorphism idea of two piles tells you that the resulting set is larger. This is false. Since there are an infinite number of elements in the two sets, the resulting set is also infinite - and the same size. (This isomorphism confounded mathematicians and philosophers for thousands of years until Georg Cantor.)

While isomorphisms might be undesirable for formal systems - Bertrand Russell certainly thought so - games distinguish themselves in that they deliberately seek to induce isomorphisms!

Think of Dungeons and Dragons. Players roll dice and assemble numbers on a piece of paper, deriving statistics and numbers. What is written down is highly abstract, the result of navigating a long series of complex rules. However, in the end, the game asks the player not to see the numbers on the paper - but a person, a character. A sentient being whose behavior they control and whose capabilities are defined by the numbers, but whose actions are their choice.

There are games that lack this appeal to isomorphism, of course. Many of the card games, such as Poker, or board games, such as Chess and Checkers, or even many sports, such as baseball and soccer. These seek to stand on their own merits, rather than insert an appeal to our imaginations. This does not make them any less of a game, it just makes their intended hook different. 

The games of key interest here are those that do rely on isomorphism. These games want players to take their decision rules and the resulting theorems and translate them into other terms. When Tephra asks a player to roll strike using their d12, it is asking the player to imagine that the use of the axioms (the d12, themselves) and the theorem (the valid move) they created with the decision rule (how tiers of strike work) not as abstract numerical constructs but as a character swinging a melee weapon at another, ready to deal damage and defend themselves. This is a complex arrangement and understanding how it works is a major step in game design itself.

Next, let’s explore the concept of isomorphism further in relation to the typographical decision rules a game chooses to have and see how that leads us to a concept of immersion.