The number line and a circle as one dimensional geometry

 This is a subtle question because the answer gets at the difference between dimension and topology.

Both a straight number line and a circle are one-dimensional spaces. At any point on either one, you can move in only two directions: "forward" or "backward" along the space. So dimension alone cannot distinguish them.

The difference is in their global structure.

The Number Line

The ordinary number line (\mathbb{R}):

• extends infinitely in both directions,
  
• has no loops,
  
• has no beginning or end, but it is unbounded,
  
• can be cut at any point and still remain connected on each side.
  

Visually:

<------------------------>

Every point corresponds to exactly one real number.

The Circle

A circle (S^1):

• is finite in extent,
  
• loops back onto itself,
  
• has no endpoints,
  
• is bounded but has no boundary.
  

Visually:

   _____

 /       \

|         |

 \_______/

If you travel far enough in one direction, you eventually return to where you started.

Why a Circle Isn't Just a Curved Line

You can bend a line into all sorts of shapes without changing its fundamental topology. For example,

~~~~~~~~~~

or

/\/\/\/\/\

are still topologically equivalent to a line.

But when you connect the ends together, something fundamentally changes. You create a closed loop.

One way mathematicians detect this is by asking:

Can the space be continuously deformed into another without cutting or gluing?

A line cannot be deformed into a circle because forming the circle requires gluing two distinct points together.

Coordinates Reveal the Difference

On the number line, positions are labeled by real numbers:

[ x \in (-\infty,\infty) ]

On a circle, positions are often labeled by an angle:

[ \theta \in [0,2\pi) ]

but with the identification

[ 0 \equiv 2\pi. ]

This identification is the key. Two different coordinate values represent the same point.

In a sense, a circle can be viewed as

[ \mathbb{R}/\mathbb{Z}, ]

which means taking the real line and declaring every integer shift to be equivalent:

[ x \sim x+n. ]

The line becomes periodic.

Connecting This to Your Earlier Question

Notice how dimension alone is not enough to characterize a space:

• A point is 0-dimensional.
  
• A line is 1-dimensional.
  
• A circle is also 1-dimensional.
  

The line and circle have the same local dimensionality, but different global topology.

Similarly, in relativity:

• Minkowski spacetime and a curved spacetime can both be 4-dimensional.
  
• What differs is the geometric and topological structure, not the dimension count.
  

So the shortest answer is:

A circle and a number line are both one-dimensional because locally they look like a line. The difference is that the number line is open and infinite, while the circle is closed and periodic—the ends are identified with each other, giving the space a different topology.

Zero dimensional space

 Yes. In both mathematics and geometry, a single point can be considered a space, and it is indeed zero-dimensional.

Here's the intuition:

• A 1-dimensional space needs one coordinate to specify a location (for example, (x) on a line).
  
• A 2-dimensional space needs two coordinates ((x,y)).
  
• A 3-dimensional space needs three ((x,y,z)).
  

A single point requires no coordinates to move within it, because there is nowhere to move. It has:

• no length,
  
• no width,
  
• no height,
  
• and no internal direction.
  

So its dimension is 0.

There is a nice progression:

Dimension	Example	Degrees of Freedom
0	Point	0
1	Line	1
2	Plane	2
3	Ordinary space	3

A useful way to think about dimension is as the number of independent directions available for motion. At a point, there are none.

This connects intriguingly with the discussion we've been having about spacetime. In Minkowski spacetime:

• An event is represented by a single point in four-dimensional spacetime ((t,x,y,z)).
  
• The event itself is 0-dimensional.
  
• The collection of all events forms the 4-dimensional spacetime manifold.
  

So when physicists say that an event (such as a flash of light occurring at a particular place and time) is a point in spacetime, they are treating it as a zero-dimensional object embedded in a four-dimensional space.

There is also a deeper geometric perspective. Just as a line can be thought of as a collection of points, some mathematicians view dimensions recursively:

• a point is a 0-dimensional space,
  
• a line is generated by moving a point,
  
• a plane by moving a line in a new orthogonal direction,
  
• ordinary space by moving a plane in a new orthogonal direction,
  
• spacetime by adding a time-like dimension to space.
  

This is very similar to the analogy you were making earlier about going from a line to a plane by introducing an orthogonal axis. The "dimension below a line" is precisely a point. The question of what would be "below a point" leads into more abstract mathematics, where some people speak informally of the empty set as having dimension (-1). That's not a physical space, but it plays an important role in topology and geometry.

So, in short: yes—a single point is a perfectly valid space, and it is the canonical example of a zero-dimensional space.

A course of study for critical game theory

 Given your background as an English instructor and your interest in meaning, myth, and systems, I'd suggest studying video game theory from three overlapping angles:

1. Games as texts (interpretation and meaning)
   
2. Games as systems (rules and mechanics)
   
3. Games as experiences (players, culture, and society)
   

A structured course of study might look like this.

Stage 1: Foundations — What Is a Game?

Read

Johan Huizinga — Homo Ludens

• Classic text arguing that play is a foundational element of culture.
  
• Useful for understanding games beyond entertainment.
  

Roger Caillois — Man, Play, and Games

• Expands Huizinga.
  
• Creates categories such as:
• Agôn (competition)
  
• Alea (chance)
  
• Mimicry (role-play)
  
• Ilinx (vertigo/disorientation)
  

Key Questions

• Why do humans play?
  
• What separates play from ordinary life?
  
• How do games create meaning?
  

Stage 2: The Ludology vs. Narratology Debate

This is one of the most important early discussions in game studies.

Read

Jesper Juul — Half-Real

• Games are both:
• real rule systems
  
• fictional worlds
  

Gonzalo Frasca

• Important advocate of ludology.
  

Janet Murray — Hamlet on the Holodeck

• Argues for understanding digital environments as narrative spaces.
  

Key Question

When we analyze a game, should we focus primarily on:

• the story?
  
• or the rules?
  

A good exercise is comparing:

• The Last of Us (story-heavy)
  
• Tetris (almost pure mechanics)
  

and asking how each produces meaning.

Stage 3: Procedural Rhetoric

This is where game studies becomes especially interesting for humanities scholars.

Read

Ian Bogost — Persuasive Games

Bogost argues that games make arguments through rules.

For example:

• A novel persuades through narrative.
  
• An essay persuades through language.
  
• A game persuades through systems.
  

A tax simulation game can make an argument about economics simply by the way its rules function.

Exercise

Play or analyze:

• Papers, Please
  
• Democracy
  
• Civilization
  

Ask:

What worldview is embedded in the rules?

Stage 4: Systems Thinking and Game Design

You mentioned Daniel Cook previously. This is where his work connects.

Read

Katie Salen & Eric Zimmerman — Rules of Play

Many consider this the foundational textbook of modern game studies.

Focus on concepts such as:

• meaningful play
  
• emergence
  
• systems
  
• feedback loops
  

Read

Daniel Cook's essays Especially:

• Skill chains
  
• Loops
  
• Value chains
  

Cook is excellent at explaining why certain game structures create engagement.

Questions

• What behaviors does the game reward?
  
• What behaviors does it punish?
  
• What is the player's long-term value chain?
  

Stage 5: Meaning, Ideology, and Criticism

This may be the most natural area for someone trained in literary analysis.

Read

Miguel Sicart — The Ethics of Computer Games

Read

Ian Bogost — How to Do Things with Videogames

Read

Aubrey Anable — Playing with Feelings

Questions

• What ethical assumptions underlie the game?
  
• What emotions are being cultivated?
  
• How does the game construct identity?
  

You can approach games much as you would novels:

• close reading
  
• rhetorical analysis
  
• ideological critique
  

but always paying attention to mechanics.

Stage 6: Culture, Community, and Players

Read

T. L. Taylor — Play Between Worlds

• Online communities
  
• MMORPG cultures
  

Read

Mia Consalvo — Cheating

• Examines player behavior and gaming culture.
  

Questions

• How do players transform games?
  
• Where does meaning reside:
• in the designer?
  
• in the system?
  
• in the community?
  

Stage 7: Advanced Theoretical Work

Once you've covered the foundations, move into:

Espen Aarseth

Cybertext

Introduces the concept of ergodic literature.

Alexander Galloway

Gaming: Essays on Algorithmic Culture

Connects games to broader questions about software and society.

McKenzie Wark

Gamer Theory

Provocative and philosophical.

Suggested Reading Order

1. Huizinga — Homo Ludens
   
2. Caillois — Man, Play, and Games
   
3. Juul — Half-Real
   
4. Salen & Zimmerman — Rules of Play
   
5. Bogost — Persuasive Games
   
6. Sicart — The Ethics of Computer Games
   
7. Daniel Cook essays
   
8. Aarseth and Galloway
   

A Literary-Theory Path

Because you teach English, another fascinating approach would be to connect game studies to theorists you may already know:

Literary Theory	Game Studies Equivalent
Reader-response	Player agency
Structuralism	Rule systems
New Criticism	Close play analysis
Rhetoric	Procedural rhetoric
Marxism	Labor, economies, progression systems
Feminist theory	Representation and player identity
Myth criticism	Archetypes, quests, hero systems

For example, you could analyze Hades, God of War, or Disco Elysium using myth criticism in much the same way you might analyze Homer or Sophocles—but with the added dimension that the player participates in constructing meaning through action.

If I were designing a semester-long independent study for an English professor, I'd center it around Huizinga → Juul → Salen & Zimmerman → Bogost → Sicart, while making close readings of Hades, Papers, Please, Civilization, Disco Elysium, and Dark Souls. Together they provide an excellent laboratory for examining narrative, systems, ideology, and player experience.