The Central Idea

A geometric theory begins with assumptions.

From those assumptions, we use valid reasoning to derive consequences.

The basic pattern is

\[ \boxed{\text{assumptions}\longrightarrow\text{logical consequences}}. \]

If the assumptions change, the resulting geometry may change as well.

Geometry Is a Structured System

A mature geometric theory usually contains several kinds of statements:

These parts play different roles and should not be confused.

Primitive Terms

A primitive term is a basic word intentionally left undefined within a formal theory.

Classical geometric systems often treat words such as

as primitive.

Their mathematical meaning is determined by how the axioms relate them, rather than by an attempt to define every term using still earlier terms.

Why Some Terms Must Remain Primitive

If every word had to be defined using earlier words, definitions would continue forever or eventually become circular.

Formal systems therefore begin with some accepted vocabulary.

Definitions are then built from those primitive terms.

Definitions

A definition assigns a precise meaning to a mathematical term.

For example, once points, lines, and incidence are available, we might define two lines to be parallel when they lie in the same plane and do not meet.

A definition does not need proof.

It establishes how a word will be used.

Axioms and Postulates

An axiom or postulate is a statement accepted as a starting assumption within the theory.

The terminology varies by context, but the logical role is the same:

\[ \boxed{\text{axioms are starting assumptions, not derived conclusions}}. \]

Theorems

A theorem is a statement proved from definitions, axioms, and previously established results.

The difference between an axiom and a theorem is therefore not necessarily how obvious the statement appears.

The difference is its role in the logical structure.

Proof

A proof is a valid chain of reasoning showing that a conclusion follows from accepted assumptions.

Schematically,

\[ A_1,A_2,\ldots,A_n \]

together with valid inference rules lead to

\[ C. \]

We then write

\[ A_1,A_2,\ldots,A_n\vdash C. \]

Logical Consequence

To say that a statement is a logical consequence of a set of assumptions means that the conclusion must hold whenever the assumptions hold.

Geometry therefore asks not merely whether a diagram looks convincing, but whether the desired conclusion follows from the stated structure.

A Familiar Example

In Euclidean geometry, the interior angles of a triangle satisfy

\[ \boxed{\alpha+\beta+\gamma=180^\circ}. \]

This is not simply a universal fact about every possible geometry.

It is a theorem that follows from Euclidean assumptions, including the Euclidean parallel structure.

Change the Assumptions

On a sphere, geodesic triangles can satisfy

\[ \alpha+\beta+\gamma>180^\circ. \]

In hyperbolic geometry,

\[ \alpha+\beta+\gamma<180^\circ. \]

These statements do not contradict the Euclidean theorem.

They belong to different geometric systems.

Conditional Mathematical Truth

A useful way to read a theorem is:

If these assumptions hold, then this conclusion follows.

Mathematics is rigorous because the assumptions are made explicit and the consequences are derived precisely.

The Parallel Postulate as a Branch Point

One of the most important examples in the history of geometry concerns parallel lines.

Given a line \(L\) and a point \(P\) not on \(L\), different geometries make different assumptions about lines through \(P\) that do not meet \(L\).

Euclidean Parallel Behavior

In Euclidean geometry, exactly one line through \(P\) is parallel to \(L\).

Symbolically, the number of such parallels is

\[ \boxed{1}. \]

Spherical Parallel Behavior

In spherical geometry, the role of lines is played by great circles.

Every pair of distinct great circles intersects.

Thus there are

\[ \boxed{0} \]

geodesic lines through \(P\) that remain disjoint from a given great circle in the ordinary spherical model.

Hyperbolic Parallel Behavior

In hyperbolic geometry, through a point outside a given line there are infinitely many lines that do not intersect the original line.

The corresponding count is

\[ \boxed{\infty}. \]

The contrast

\[ 0,\qquad1,\qquad\infty \]

dramatically illustrates how one structural assumption can organize entire families of geometry.

Models

A model is a concrete mathematical structure in which the axioms of a theory are satisfied.

Models help us interpret abstract assumptions.

They also provide a powerful way to investigate consistency, independence, and possible behavior.

The Euclidean Plane as a Model

Ordinary coordinate space

\[ \mathbb{R}^2 \]

with its usual points and straight lines provides a standard model of Euclidean plane geometry.

Algebraic coordinates make geometric claims testable through equations.

The Sphere as a Model

A sphere with great circles treated as lines provides a model of spherical geometry.

In that model:

Models of Hyperbolic Geometry

Hyperbolic geometry can be represented using several models.

Examples include:

These models look different but encode the same underlying hyperbolic geometry.

A Model Is Not the Geometry Itself

The Poincaré disk is drawn inside an ordinary Euclidean disk.

That does not mean hyperbolic distance is ordinary Euclidean distance in the picture.

A model provides a representation together with rules for interpreting geometric objects and measurements.

Representations Can Distort Appearance

In one model, a hyperbolic line may appear as a circular arc.

In another, the same abstract kind of line may appear as a Euclidean chord.

What is invariant is the underlying geometric relationship, not the visual form imposed by the representation.

Projective Geometry Changes What Matters

Projective geometry deemphasizes ordinary metric quantities such as Euclidean lengths and angles.

Instead, it emphasizes incidence:

Changing the assumptions changes the natural invariants.

Metric Geometry Changes the Meaning of Distance

In metric geometry, the central choice is a distance function

\[ d(x,y). \]

The metric must satisfy specified axioms.

Different metrics on the same underlying set can create different geometric behavior.

One Set, Different Geometries

Consider the coordinate plane

\[ \mathbb{R}^2. \]

With Euclidean distance,

\[ d_2((x_1,y_1),(x_2,y_2)) = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. \]

With taxicab distance,

\[ d_1((x_1,y_1),(x_2,y_2)) = |x_2-x_1|+|y_2-y_1|. \]

The point set is unchanged, but the geometry differs because distance has changed.

Metric Balls Reveal the Difference

In the Euclidean metric, a set of points at fixed distance \(r\) from a center forms an ordinary circle.

In the taxicab metric, the same distance condition produces a diamond-shaped boundary in ordinary Cartesian coordinates.

The definition

\[ d(P,C)=r \]

stays the same while the metric changes.

Finite Geometry Changes the Underlying Universe

Finite geometry demonstrates another possibility.

Instead of infinitely many points and lines, the entire incidence system may be finite.

Geometry can therefore be organized around a finite collection of objects satisfying exact axioms.

The Fano Plane as an Axiomatic Model

The Fano plane contains seven points and seven lines.

Every two distinct points lie on exactly one line.

Every two distinct lines meet in exactly one point.

These incidence rules, not the particular drawing, define the projective plane of order \(2\).

Manifolds Change the Scale of Description

Manifold geometry allows a space to behave one way locally and another way globally.

A sphere is locally modeled on

\[ \mathbb{R}^2 \]

even though it is not globally a Euclidean plane.

The assumptions concern local coordinate structure rather than a single global flat representation.

Definitions Determine the Objects of Study

Mathematical words do not carry all of their everyday meaning into a formal theory.

A "line" in one geometry may not be represented by a Euclidean straight mark in another model.

What matters is the role specified by the theory.

Geodesic as an Example

In Euclidean space, geodesics are ordinary straight lines.

On a sphere, geodesics are great circles.

On a curved manifold, geodesics are determined by the local metric.

The concept is more general than its most familiar Euclidean example.

Derived Properties

Once assumptions are fixed, many familiar properties become derived rather than assumed.

Examples include:

Dependence on Assumptions

When proving a theorem, it is useful to ask:

Which assumptions were actually used?

A theorem may remain valid under weaker assumptions than the full theory.

Identifying the minimum required assumptions reveals the logical structure more clearly.

Independence

An axiom is independent of the other axioms when it cannot be proved from them.

One common way to demonstrate independence is to construct models in which all the other axioms hold but the questioned axiom behaves differently.

The Parallel Postulate and Independence

For centuries, mathematicians attempted to derive the Euclidean parallel postulate from simpler-looking assumptions.

The eventual construction of non-Euclidean geometries showed that changing the parallel assumption can produce coherent alternative systems.

This revealed that Euclidean parallel behavior was not forced by the other basic geometric assumptions in the way many had expected.

Consistency

A formal theory is consistent if it does not allow both a statement and its negation to be derived.

Informally, a consistent system does not logically contradict itself.

Models and Relative Consistency

Models can provide evidence of relative consistency.

If a new geometric theory can be modeled inside an accepted mathematical framework, then a contradiction in the new theory would produce a contradiction in the framework used to construct the model.

This was historically important in establishing the legitimacy of non-Euclidean geometry.

Consistency Is Not the Same as Physical Accuracy

A geometry can be logically consistent without being the best model of a particular physical situation.

Mathematical consistency asks whether the theory coheres logically.

Scientific adequacy asks whether a model describes observations well.

These are related but different questions.

Geometry and the Physical World

Mathematical geometry supplies possible structures.

Physics decides which structures best model particular physical phenomena.

For small-scale construction on nearly flat ground, Euclidean geometry is often excellent.

For global navigation, spherical or ellipsoidal geometry becomes more appropriate.

For gravitation and spacetime, differential geometry provides a more suitable framework.

Approximation and Scale

Different geometries may be useful at different scales.

A small patch of Earth's surface can often be approximated as flat.

Over intercontinental distances, that approximation becomes inadequate.

The choice of geometry is therefore partly a modeling decision.

Euclidean Geometry as a Local Approximation

Smooth curved spaces often resemble Euclidean space at sufficiently small scales.

This is why tangent planes and local coordinates are so useful.

Euclidean methods can remain approximately correct locally even when the global geometry is curved.

Counterexamples

A counterexample disproves a universal claim by providing one valid case in which the claim fails.

In geometry, alternative models are especially powerful sources of counterexamples to overgeneralized Euclidean intuition.

Example: Overgeneralizing the Triangle Sum

Claim:

"Every triangle has angle sum \(180^\circ\)."

A spherical triangle with three right angles has

\[ 90^\circ+90^\circ+90^\circ=270^\circ. \]

Therefore the universal statement is false unless the geometric setting is restricted appropriately.

Example: Overgeneralizing Unique Shortest Paths

In Euclidean space, two distinct points determine a unique straight segment of minimum length.

In taxicab geometry, many distinct shortest routes can join the same two points.

Thus uniqueness of shortest paths depends on the geometry.

Example: Overgeneralizing Parallel Lines

In Euclidean geometry, two distinct lines with equal slope are parallel.

In projective geometry, parallel affine lines are completed by a common point at infinity.

The projective theory changes the underlying incidence assumptions.

Equivalent Formulations

Sometimes different-looking statements are logically equivalent within a given theory.

That means each statement can be derived from the other together with the remaining assumptions.

Equivalent formulations can reveal hidden connections between geometric properties.

Why Equivalent Axioms Matter

A theory may be presented using different axiom systems.

If two axiom systems generate the same consequences, they may describe the same mathematical structure from different starting points.

One formulation may be better for teaching, another for proof, and another for computation.

Minimal Assumptions

Mathematicians often seek axiom systems that are

This is a logical design problem as much as a geometric one.

Completeness

In an informal geometric sense, one may ask whether the assumptions are sufficient to settle the kinds of questions the theory intends to address.

In formal logic, completeness has more technical meanings.

At this level, the important distinction is simply that a consistent axiom system need not automatically answer every imaginable statement.

Undefined Does Not Mean Meaningless

A primitive term can be mathematically useful even when it has no explicit definition inside the theory.

Its behavior is constrained by the axioms.

This is similar to how an algebraic variable can be manipulated according to rules before a particular numerical value is assigned.

Diagrams as Evidence, Not Proof

Diagrams are valuable for intuition.

But a diagram can hide special cases, distort measurements, or suggest properties that were never assumed.

A proof must rest on definitions, assumptions, and valid reasoning rather than appearance alone.

The Danger of Imported Assumptions

A common mistake is to use a familiar Euclidean property inside a non-Euclidean setting without justification.

For example, one cannot assume

\[ \alpha+\beta+\gamma=180^\circ \]

in a spherical triangle merely because the figure is called a triangle.

The governing geometry must be identified first.

Geometry as a Choice of Structure

Different geometries emphasize different structures.

Invariants

An invariant is a property preserved under the class of transformations or equivalences relevant to a theory.

Different geometries preserve different information.

Understanding a geometry often means understanding what it treats as invariant.

Euclidean Invariants

Euclidean rigid motions preserve

These preserved quantities help define congruence.

Similarity Invariants

Similarity transformations preserve

Absolute length is not preserved.

Affine Invariants

Affine transformations preserve properties such as

Ordinary lengths and angles need not be preserved.

Projective Invariants

Projective transformations preserve

Parallelism, ordinary length, and ordinary angle measure are generally not projective invariants.

A Hierarchy of Structure

One useful way to organize geometric theories is by asking how much structure is retained.

Moving from Euclidean to similarity to affine to projective geometry progressively discards some metric information while retaining more basic incidence structure.

This is not a ranking from better to worse.

Each theory is adapted to different questions.

Geometry as Modeling

A geometry is useful when its assumptions capture the relationships that matter in a problem.

Choosing a geometry is therefore similar to choosing any mathematical model:

  1. identify the objects;
  2. identify the important relationships;
  3. choose assumptions representing those relationships;
  4. derive consequences;
  5. compare those consequences with the problem being modeled.

Example: Local Construction

Suppose a builder lays out a rectangular foundation over a small site.

Euclidean geometry is an excellent model because Earth curvature is negligible at that scale.

The useful assumptions include flatness, straight-line distances, and right angles.

Example: Long-Distance Navigation

For a route across the globe, a flat-plane model can introduce significant distortion.

A spherical or ellipsoidal model better captures the geometry.

Shortest surface paths then follow geodesic principles rather than ordinary straight segments on a flat map.

Example: City-Block Travel

If movement is constrained to an orthogonal street grid, taxicab distance may better represent travel than Euclidean distance.

The physical locations remain the same.

The metric changes because the allowed movement structure changes.

Example: Computer Vision

In perspective imaging, projective geometry is often more natural than Euclidean geometry.

Camera projection does not preserve ordinary distances or angles, but it does preserve important projective relationships.

Choosing the correct geometry simplifies the problem.

Example: Finite Networks

When the objects of interest are finite and incidence-based, finite geometry can provide useful structure.

Such models arise in coding theory, combinatorial designs, and related discrete systems.

Example: Curved State Spaces

In robotics or physics, the possible states of a system may form a manifold.

Ordinary Cartesian coordinates may work only locally.

The correct geometric model must account for the global structure of the state space.

Assumptions Should Be Declared

A well-posed geometric problem should make its setting clear.

Useful questions include:

Hidden Assumptions Cause Errors

Many incorrect solutions are not algebra mistakes.

They arise because an unstated assumption was imported from a familiar setting.

A reliable habit is to identify the geometric framework before applying a theorem.

Theorem Conditions Matter

A theorem should never be separated from its hypotheses.

Instead of remembering only

\[ a^2+b^2=c^2, \]

remember that the Pythagorean theorem concerns a right triangle in Euclidean geometry.

The conditions determine when the conclusion is valid.

Definitions Can Also Change

Sometimes a new geometry keeps familiar words but generalizes their definitions.

For example, a geodesic generalizes the idea of a straight path.

A metric generalizes the idea of distance.

A manifold generalizes familiar curves and surfaces to arbitrary dimension.

Abstraction Preserves Structure

Mathematical abstraction does not merely make ideas less concrete.

It isolates the structure shared by many examples.

Once that structure is identified, a theorem can apply to every model satisfying the assumptions.

Why Alternative Geometries Matter

Alternative geometries teach a general mathematical lesson:

Familiar conclusions should be traced back to the assumptions that make them true.

This habit strengthens reasoning far beyond geometry.

From Geometry to Mathematics More Broadly

The same structure appears throughout mathematics:

\[ \text{objects} + \text{definitions} + \text{axioms} + \text{inference} \longrightarrow \text{theory}. \]

Algebra, analysis, probability, topology, and other fields all rely on explicit structures and logically controlled conclusions.

Common Error: Treating Axioms as Proven Facts

An axiom is not a theorem whose proof has been omitted.

It is a chosen starting assumption within the system.

Its role is foundational rather than derived.

Common Error: Treating Definitions as Theorems

A definition establishes meaning.

It does not assert a discovered consequence that requires proof.

For example, if a term is defined by a certain condition, applying that condition is unpacking the definition, not proving a new theorem.

Common Error: Assuming Different Geometries Contradict One Another

The statements

\[ \alpha+\beta+\gamma=180^\circ, \] \[ \alpha+\beta+\gamma>180^\circ, \]

and

\[ \alpha+\beta+\gamma<180^\circ \]

can all be valid because they refer to different geometric assumptions.

Common Error: Confusing a Model with Physical Reality

A mathematical model is a structured representation.

It may approximate or describe a physical system extremely well, but the mathematical object and the physical object are not identical.

Common Error: Assuming the Picture Defines the Geometry

A projective line may be drawn as a circle in a finite diagram.

A hyperbolic line may be drawn as an arc in one model.

A map of a sphere may flatten curved geography.

The representation must be interpreted according to the rules of the model.

Common Error: Ignoring Hypotheses

A theorem cannot be applied merely because its conclusion looks useful.

The hypotheses must first be checked.

This is especially important when moving among Euclidean, spherical, hyperbolic, affine, projective, and metric settings.

Common Error: Assuming More Structure Than Was Given

If a problem specifies only an incidence geometry, ordinary distance may not even be defined.

If a problem specifies only a smooth manifold, a Riemannian metric may not have been chosen.

Use only the structure actually provided.

A Reliable Procedure for Reading a Geometric Theory

  1. Identify the primitive objects.
  2. Identify the definitions.
  3. List the axioms or structural assumptions.
  4. Determine which transformations or relationships matter.
  5. Identify the quantities or properties that are invariant.
  6. Separate assumptions from derived theorems.
  7. Use valid reasoning to derive consequences.
  8. Test broad claims against alternative models or counterexamples.

A Reliable Procedure for Comparing Two Geometries

  1. Compare their underlying objects.
  2. Compare what each theory calls a line or geodesic.
  3. Compare their parallel rules.
  4. Compare their distance structures.
  5. Compare their curvature assumptions.
  6. Compare their invariant properties.
  7. Compare representative theorems such as triangle angle sums.
  8. Identify which assumptions explain the differences.

A Reliable Procedure for Choosing a Geometry in Modeling

  1. Identify the physical or abstract objects being modeled.
  2. Determine whether distance, angle, incidence, curvature, or topology is central.
  3. Determine the relevant scale.
  4. Choose the simplest geometry that preserves the important structure.
  5. State the assumptions explicitly.
  6. Derive predictions or consequences.
  7. Check whether the model remains appropriate across the intended domain.

Example: Classifying a Triangle-Sum Claim

Suppose someone states:

"The angles of every triangle sum to \(180^\circ\)."

A careful response is:

This is true for ordinary Euclidean triangles, but it is not a theorem of every geometry.

The correct classification is therefore

\[ \boxed{\text{a Euclidean theorem, not a geometry-independent definition}}. \]

Example: What Changed?

Compare two systems:

System A: exactly one parallel through an external point.

System B: infinitely many nonintersecting lines through an external point.

The primary changed assumption concerns

\[ \boxed{\text{parallel structure}}. \]

The resulting theories are Euclidean and hyperbolic in character.

Example: Same Points, Different Metric

Let

\[ A=(0,0),\qquad B=(3,4). \]

Euclidean distance:

\[ d_2(A,B)=5. \]

Taxicab distance:

\[ d_1(A,B)=7. \]

The coordinates are unchanged.

The differing conclusion comes entirely from a different metric assumption.

Example: Same Local Dimension, Different Global Space

A sphere and a torus are both two-dimensional manifolds.

Locally, both resemble

\[ \mathbb{R}^2. \]

Globally, however, they have different topological structure.

Thus

\[ \boxed{\text{same local dimension}\not\Rightarrow\text{same global geometry}}. \]

Example: A Finite Projective Plane

Suppose a geometry satisfies:

These assumptions describe the incidence pattern of the Fano plane.

Its seven-point structure is a consequence of the finite projective-plane axioms with order \(2\).

Example: Detecting an Imported Assumption

A student reasons:

"These two geodesics cannot meet twice because two straight lines meet at most once."

On a sphere, two distinct great circles meet at a pair of antipodal points.

The student imported a Euclidean line theorem into spherical geometry.

Practice 1: Definition, Axiom, or Theorem?

Classify each statement by role:

  1. A line segment joining the center of a circle to a point on the circle is called a radius.
  2. Through a point not on a given line, exactly one parallel line exists.
  3. The angle sum of a Euclidean triangle is \(180^\circ\).

Practice 2: Changing Parallel Behavior

Match each number of parallels through an external point to the familiar geometry:

\[ 0,\qquad1,\qquad\infty. \]

Use spherical, Euclidean, and hyperbolic geometry.

Practice 3: Model or Theory?

The Poincaré disk is drawn as an ordinary Euclidean disk.

Does this mean hyperbolic distance inside the model is ordinary Euclidean distance?

Practice 4: Hidden Assumption

A student claims that every triangle must have angle sum \(180^\circ\) because "that is what triangles do."

Identify the hidden assumption.

Practice 5: Metric Choice

In \(\mathbb{R}^2\), let

\[ A=(1,1),\qquad B=(5,4). \]

Find the Euclidean and taxicab distances.

Practice 6: Counterexample

Give a geometric setting that provides a counterexample to the claim:

"Two points always have exactly one shortest path between them."

Practice 7: Local versus Global

Explain why the fact that both a sphere and torus are locally two-dimensional does not imply that they are globally equivalent.

Practice 8: Incidence versus Metric Structure

Suppose a finite geometry specifies only points, lines, and incidence.

Are ordinary Euclidean lengths automatically meaningful?

Practice 9: Independence

In broad terms, what kind of evidence can show that an axiom is independent of the others?

Practice 10: Modeling Choice

Choose the most natural starting geometry for each situation:

  1. laying out a small rectangular room;
  2. planning the shortest route over Earth's surface between distant cities;
  3. studying perspective images from a camera;
  4. travel constrained to an orthogonal street grid.

Practice Answers

Practice 1

  1. The statement introducing the word radius is a definition.
  2. The unique-parallel statement is used as an axiom or postulate in an affine Euclidean-style system.
  3. The \(180^\circ\) triangle-angle statement is a theorem of Euclidean geometry.

Practice 2

\[ \boxed{ 0\rightarrow\text{spherical}, \qquad 1\rightarrow\text{Euclidean}, \qquad \infty\rightarrow\text{hyperbolic} }. \]

Practice 3

No.

The Euclidean disk is a representation of the hyperbolic model. Hyperbolic distance is defined by the model's own metric, not by ordinary Euclidean distance in the drawing.

Practice 4

The student has silently assumed

\[ \boxed{\text{Euclidean geometry}}. \]

Other geometries allow different triangle angle sums.

Practice 5

Coordinate differences are

\[ \Delta x=4,\qquad\Delta y=3. \]

Euclidean distance:

\[ d_2=\sqrt{4^2+3^2}=5. \]

Taxicab distance:

\[ d_1=|4|+|3|=7. \]

Therefore,

\[ \boxed{d_2=5,\qquad d_1=7}. \]

Practice 6

Taxicab geometry provides a counterexample.

Many different monotone grid-like routes can have the same minimum taxicab length between two points.

Practice 7

Local dimension records how many coordinates are needed near each point.

Global topology records the structure of the entire space.

A sphere and torus both need two local coordinates, but the torus has a different global connectivity and hole structure.

Practice 8

No.

Incidence alone specifies which points lie on which lines.

A metric must be supplied separately before numerical distances become part of the geometry.

Practice 9

One strategy is to construct models satisfying the other axioms in which the questioned axiom takes different truth values.

If both kinds of models exist, the axiom cannot be forced by the remaining assumptions.

Practice 10

  1. small room: Euclidean geometry;
  2. distant Earth route: spherical or ellipsoidal geometry;
  3. camera perspective: projective geometry;
  4. orthogonal street grid: taxicab metric geometry.

What to Remember

Section Synthesis

The study of non-Euclidean and alternative geometries reveals a broader picture of mathematics.

Spherical and hyperbolic geometry showed that curvature changes geodesics, parallelism, distance, and triangle angle sums.

Projective geometry showed that incidence can remain meaningful even when ordinary lengths and angles are discarded.

Alternative metrics showed that the same set of points can support different geometries depending on how distance is defined.

Finite geometry showed that a geometric universe can contain only finitely many points and lines while retaining exact structural laws.

Manifolds showed how local Euclidean structure can coexist with complicated global form.

Together, these examples demonstrate that geometry is best understood not as one fixed picture of space, but as the systematic study of structures generated by explicit assumptions.

Final Perspective

The deepest habit developed in this section is not memorizing which geometry has which formula.

It is learning to ask:

Once those questions become routine, alternative geometries stop looking like exceptions to geometry.

They become what they are: different coherent consequences of different mathematical structures.