New model of non-Euclidean plane

TL;DR

The paper introduces a novel non-Euclidean plane model based on hyperreal numbers, illustrating key geometric axioms and differences from hyperbolic geometry with accessible educational benefits.

Contribution

It presents a new non-Euclidean plane model within hyperreal numbers, enabling representation of various geometric axioms and distinctions from hyperbolic planes.

Findings

Model represents triangle angle sum as π.

Allows visualization of parallel axiom negations.

Highlights differences between non-Euclidean and hyperbolic planes.

Abstract

We present a new model of a non-Euclidean plane, in which angles in a triangle sum up to $\pi$. It is a subspace of the Cartesian plane over the field of hyperreal numbers $\mathbb{R}^*$. The model enables one to represent the negation of equivalent versions of the parallel axiom, such as the existence of the circumcircle of a triangle, and Wallis' or Lagendre's axioms, as well as the difference between non-Euclidean and hyperbolic planes. The model has unique educational advantages as expounding its crucial ideas requires only the basics of Cartesian geometry and non-Archimedean fields.

Equations

$$\mathbb{L}=\bigcup _{r\in \mathbb{R}}r+\Omega .$$

$$f^{\ast }([(r_n)])=[(f(r_1),f(r_2),...)].$$

$${\sqrt{[(r_n)]}}^{\ast }=[(\sqrt{r_n})]=[(\sqrt{r_1},\sqrt{r_2},...)],\text{mbox}for[(r_j)]>0.$$

$${\sin }^{\ast }[(r_n)]=[(\sin r_1,\sin r_2,...)],{\cos }^{\ast }[(r_n)]=[(\cos r_1,\cos r_2,...)].$$

$$({\sin }^{\ast }x{)}^2+({\cos }^{\ast }x{)}^2=1.$$

$$x^{\prime }=\frac{c_x}{\sqrt{c_x^2+c_y^2}}x-\frac{c_y}{\sqrt{c_x^2+c_y^2}}y,$$

$$y^{\prime }=\frac{c_y}{\sqrt{c_x^2+c_y^2}}x+\frac{c_x}{\sqrt{c_x^2+c_y^2}}y.$$

Full text

The model has unique educational advantages as expounding its crucial ideas requires only the basics of Cartesian geometry and non-Archimedean fields.

Keywords: Parallel postulate, semi-Euclidean plane, hyperbolic geometry, hyperreal numbers

1. 1 Introduction

2. 2 Hilbert axioms for Euclidean and hyperbolic geometry

3. 3 Semi-Euclidean plane

4. 3.1 The Cartesian plane over the field of hyperreal numbers

5. 3.2 Circumcircle of a triangle, Wallis’ and Lagendre’s axioms

6. 3.3 Trigonometry

7. 4 Limiting parallel rays

1. Introduction

Klein and Poincaré disks are classical models of non-Euclidean geometry. Both consist of a fixed circle in the Euclidean plane, say $\Gamma$, representing the plane. In the Klein disk, chords of $\Gamma$ are straight lines; in the Poincaré disk, straight lines are diameters of $\Gamma$ or arcs of circles orthogonal to $\Gamma$ (Fig. 1).

In the Poincaré model, an angle between intersecting circles is the Euclidean angle between tangents to these lines drawn at their intersection point (Fig. 2). In the Klein disc, an angle between intersecting straight lines is retrieved from the Poincaré model as presented in Fig. 3: for lines $n,m$, we draw circles orthogonal to $\Gamma$ and determine the angle between them.

Standard models of non-Euclidean planes, thus, involve a non-Euclidean representation of straight lines (Poincaré) or angles (Klein). We present a model in which both straight lines and angles are Euclidean as it takes the concept of a straight line and angle from the Cartesian plane over an ordered field.

A Euclidean plane is a Hilbert plane satisfying the parallel axiom and circle-circle intersection axiom. The Cartesian plane over an ordered field closed under the square root operation (i.e., over a Euclidean field) provides a model of the Euclidean plane [7, 153]. Section 3 of our paper introduces a non-Archimedean field of hyperreals $({\mathbb{R}}^{*},+,\cdot,0,1,<)$. Since it is a Euclidean field, the Cartesian plane ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$ is a model of Euclidean geometry. We show that its subspace $\mathbb{L}\times\mathbb{L}$, where $(\mathbb{L},+,\cdot,0,1)$ is the ring of limited hyperreal numbers, is a semi-Euclidean plane, that is, it satisfies Hilbert’s axioms of the so-called absolute geometry, does not satisfy the parallel and Archimedean axioms, but angles in any triangle sum up to $\pi$ (i.e., two right angles).

2. Hilbert axioms for Euclidean and hyperbolic geometry

Hilbert’s Grundlagen der Geometrie, from [8] to [10] got eleven editions. [7] includes its modern version adjusted to educational practice. Hilbert axioms, as presented therein, differ from the original only in applying modern symbols.111[6, 597–602] provides a concise account of Hilbert axioms; see also [3].

Hilbert grouped his axioms due to primitive concepts, point, straight line, plane, and the relation of betweenness, congruence of line segments, and angles. The paper will discuss three of them.

Archimedes’ axiom

Given line segments $AB$ and $CD$, there is a natural number $n$ such that $n$ copies of $AB$ added together will be greater than $CD$.

Parallel axiom

For each point $A$ and each line $l$, there is at most one line containing $A$ that is parallel to $l$.

Hyperbolic axiom of parallels

If $l$ is any line and $A$ a point not on the line, then there exist through $A$ two rays $AX^{\rightarrow}$ and $AY^{\rightarrow}$ which do not form one and the same line and do not intersect the line $l$, while every ray emanating from $A$ that lies in the angle formed by $AX^{\rightarrow}$ and $AY^{\rightarrow}$ does intersect $l$.222See [9, 136], [6, 259], [7, 374].

Absolute (neutral) geometry consists of axioms of incidence, betweenness, congruence of line segments and angles. One can also view it as a set of propositions Elements I.1–28.

Generally, parallel lines are not intersecting lines. In the neutral geometry, the transportation of angles enables one to construct a line through $A$ that does not meet $l$. Hilbert’s parallel axiom translates, thus, into the claim: There is exactly one line containing $A$ that is parallel to $l$.333[3, 75–77] discusses a relationship between Hilbert’s and Euclid’s parallel axioms. In hyperbolic geometry, next to not intersecting lines, one considers limiting rays. Thus, Euclid’s parallel axiom requires one line through $A$, not meeting $l$, its negation – at least two lines through $A$ not meeting $l$, the hyperbolic axiom of parallels requires exactly two limiting rays originating in $A$, not meeting $l$. In what follows, we will present a non-Euclidean plane that is not hyperbolic.

3. Semi-Euclidean plane

In this section, we present a model of a semi-Euclidean plane, i.e., a plane in which angles in a triangle sum up to $\pi$ yet the parallel postulate fails. [7, 311], introduces that term, but the very idea originates in [4, § 9]. Dehn built such a model owing to a non-Archimedean Pythagorean field introduced in [8, § 12]; yet, it was a non-Euclidean field.444See also [7, § 18]. Example 18.4.3 expounds on Dehn’s model. We employ the Euclidean field of hyperreal numbers. To elaborate, let us start by introducing these numbers.

3.1. The Cartesian plane over the field of hyperreal numbers

An ordered field $({\mathbb{F}},+,\cdot,0,1,<)$ is a  commutative field together with a total order that is compatible with sums and products. In such a field, one can define the following subsets of ${\mathbb{F}}$:

1. $\mathbb{L}=\{x\in{\mathbb{F}}:(\exists n\in{\mathbb{N}})(|x|<n)\}$,
2. $\Psi=\{x\in{\mathbb{F}}:(\forall n\in{\mathbb{N}})(|x|>n)\}$,
3. $\Omega=\{x\in{\mathbb{F}}:(\forall n\in{\mathbb{N}})(|x|<\tfrac{1}{n})\}$.

They are called limited, infinite, and infinitely small numbers, respectively. Here are some relationships helpful to pursue our arguments.

1. $(\forall x,y\in\Omega)(x+y\in\Omega,xy\in\Omega)$,
2. $(\forall x,y\in\mathbb{L})(x+y\in\mathbb{L},xy\in\mathbb{L})$,
3. $(\forall x\in\Omega)(\forall y\in\mathbb{L})(xy\in\Omega)$,
4. $(\forall x\neq 0)(x\in\Omega\Leftrightarrow\ x^{-1}\in\Psi)$.

We can also encode these rules by terms such as $\Omega+\Omega\subset\Omega$, or $\Omega\Omega\subset\Omega$, etc.

To clarify our account, let us observe that the following equality $\Omega=\{0\}$ is a version of the well-known Archimedean axiom.

Since real numbers form the biggest Archimedean field, every field extension of $({\mathbb{R}},+,\cdot,0,1,<)$ includes positive infinitesimals. The standard way of extending real numbers employs formal power series, below we sketch another method.

Let $\mathcal{U}$ be a non-principal ultrafilter on ${\mathbb{N}}$. The set of hyperreals is defined as a reduced product ${\mathbb{R}}^{*}={\mathbb{R}}^{{\mathbb{N}}}/{\mathcal{U}}$. Sums, products, and the order are introduced pointwise. The field of hyperreals $({\mathbb{R}}^{*},+,\cdot,0,1,<)$ extends real numbers, hence, includes infinitesimals and infinite numbers [1], [2]. Fig. 4 represents in a schematized way a relationship between ${\mathbb{R}}$ and ${\mathbb{R}}^{*}$, as well as between $\mathbb{L}$, $\Psi$, and $\Omega$.

To get an algebraic insight, let us note that limited numbers form an ordered ring, while infinitesimals are its maximal ideal. Due to the so-called standard part theorem, one can show that the quotient ring is isomorphic to the field of real numbers. In consequence, the set $\mathbb{L}$ finds the following representation

[TABLE]

Although the below diagrams spot on the origin of the coordinate system, one can reiterate our arguments taking into account any point $(r,s)$, where $r,s\in{\mathbb{R}}$, and its infinitesimal neighborhood, i.e., $(r,s)+\Omega\times\Omega$, in short $(r,s)+\Omega^{2}$.

In section § 3.3 below, we show how to introduce trigonometric functions and other counterparts of real maps, such as the square root. Thus, the field of hyperreals is a Euclidean field.

Due to the proposition [7, 16.2], the Cartesian plane over the field of hyperreals is a model of Euclidean plane, with straight lines and circles given by equations $ax+by+c=0$, $(x-a)^{2}+(y-b)^{2}=r^{2}$, where $a,b,c,r\in{\mathbb{R}}^{*}$ and due to the equation of a straight line, parameters $a,b$ have to satisfy condition $a^{2}+b^{2}\neq 0$; angles between straight lines are defined as in the Cartesian plane over the field of real numbers. Specifically, on the plane ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$, angles in triangles sum up to $\pi$. Parallel lines are of the form $y=mx+b$ and $y=mx+c$, while a perpendicular to the line $y=mx+b$ is given by the formula $y=-\frac{1}{m}x+d$.

Now, let us take a subspace $\mathbb{L}\times\mathbb{L}$ of the plane ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$. In that plane, circles are defined by analogous formula, namely $(x-a)^{2}+(y-b)^{2}=r^{2}$, where $a,b,c,r\in\mathbb{L}$, while every line in $\mathbb{L}\times\mathbb{L}$ is of the form $l\cap\mathbb{L}\times\mathbb{L}$, where $l$ is a line in ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$. Since we want plane $\mathbb{L}\times\mathbb{L}$ include lines such as $y_{1}=\varepsilon x$, where $\varepsilon\in\Omega$, it also has to include the perpendicular $y_{2}=\frac{-1}{\varepsilon}x$, but $\frac{-1}{\varepsilon}\notin\mathbb{L}$. Formula $l\cap\mathbb{L}\times\mathbb{L}$, where $l=ax+by+c$ and $a,b,c\in{\mathbb{R}}^{*}$ guarantees the existence of the straight line $y_{2}$ in $\mathbb{L}\times\mathbb{L}$. Finally, the interpretation of an angle is the same as in the model ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$.

Explicit checking shows that the model characterized above satisfies Hilbert axioms of absolute plane geometry plus the circle-circle and line-circle axioms; see [7, §14–17].

With regard to parallel lines, let us consider the horizontal line $y\equiv 1$ and two specific lines through $(0,0)$, namely $y_{1}=\varepsilon x,y_{2}=\delta x$, where $\varepsilon,\delta\in\Omega$; see Fig. 6. Since $\Omega\mathbb{L}\subset\Omega$, the following inclusions hold $y_{1},y_{2}\subset\mathbb{L}\times\Omega$. In other words, values of maps $y_{1}(x),y_{2}(x)$ are infinitesimals, given that $x\in\mathbb{L}$. The same obtains for any line of the form $y=\mu x$, with $\mu\in\Omega$. Since there are infinitely many infinitesimals, there are infinitely many lines through $(0,0)$ not intersecting the horizontal line $y\equiv 1$.

Since every triangle in $\mathbb{L}\times\mathbb{L}$ is a triangle in ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$, it follows that angles in a triangle on the plane $\mathbb{L}\times\mathbb{L}$ sum up to $\pi$ (Fig. 7).

3.2. Circumcircle of a triangle, Wallis’ and Lagendre’s axioms

In our model, one can easily find counterexamples for various versions of the parallel axiom. Below, we discuss three of them.

In the Elements, Book IV, proposition 5, Euclid requires to circumscribe a circle about a given triangle. To this end, he assumes that perpendicular bisectors of two sides of a triangle meet. Indeed, the intersection point is the center of the circle circumscribing a triangle. The existence of such a circumcircle is an equivalent version of the parallel axiom. Below we show it does not hold in the plane $\mathbb{L}\times\mathbb{L}$.

Let us take the line $y\equiv-\varepsilon$ and points $A=(-1,-\varepsilon)$, $B=(1,-\varepsilon)$ on it. A line through the point $C=(0,0)$ and $A$ has the equation $y=\varepsilon x$. Perpendicular bisectors of the sides $AB$ and $AC$ have the equations $x=0$ and $2x+2\varepsilon y+\varepsilon^{2}+1=0$, respectively. They meet at the point $\left(0,-\varepsilon-\frac{1}{\varepsilon}\right)$. However, $\left(0,-\varepsilon-\frac{1}{\varepsilon}\right)\notin\mathbb{L}\times\mathbb{L}$. Fig. 8 depicts three perpendiculars to the sides of the triangle $ABC$.

The so-called Wallis axiom reads: Given any triangle $\triangle ABC$ and given any segment $DE$, there exists a triangle $\triangle DEF$ having $DE$ as one of his sides such that $\triangle ABC\sim\triangle DEF$ [6, 216].

It is yet another equivalent version of the parallel axiom [6, 216–217].

Let $ABC$ be a triangle with vertices $O$, $(\varepsilon,0)$, $(1,0)$, and take $DE=[0,1]$. Then, the triangle $DEF$ equiangular with the triangle $ABC$ has the vertex $F=(0,\tfrac{1}{\varepsilon})$ outside the plane $\mathbb{L}\times\mathbb{L}$.

Finally, let us take Legender’s axiom that reads: For any angle and any point $A$ in the interior of that angle, there exists a line through $A$ and not through that angle vertex which intersects both sides of the angle [6, 223].

Lagendre’s axiom is equivalent to the parallel axiom, given the Archimedean axiom [7, 324].

To show it fails in our model, take $A=(0,1)$, and an angle with arms made by rays $y_{1}\equiv 0$ and $y_{2}=\tan(\pi-\varepsilon)x$ emanating from $O$. Now, any line cutting $y_{1}$ does not meet $y_{2}$. Moreover, the straight line $y_{3}\equiv 1$ does not meet any arm of the angle.

3.3. Trigonometry

Let $f$ be a real map, i.e. $f:{\mathbb{R}}\mapsto{\mathbb{R}}$. Its extension to a map on ${\mathbb{R}}^{*},$ $f^{*}:{\mathbb{R}}^{*}\mapsto{\mathbb{R}}^{*}$, is defined by

[TABLE]

If $r\in{\mathbb{R}}$, then $f^{*}(r)=[(f(r),f(r),...)]$. Since we identify real number $r$ with hyperreal $[(r,r,...)]$, the equality $f^{*}([(r,r,...)])=f(r)$ obtains, meaning, $f^{*}$ extends $f$, $f^{*}_{|{\mathbb{R}}}=f$.

Putting $f=\sqrt{\ \ }$ in definition (3.1), we get

[TABLE]

Similarity, under the definition (3.1), we have

[TABLE]

Since for every $n$ the identity $\sin^{2}r_{n}+\cos^{2}r_{n}=1$ holds, we have

[TABLE]

Similarly, every trigonometric identity translates into an identity involving the maps $\sin^{*}$ and $\cos^{*}$, $\tan^{*}$ and $\cot^{*}$. Interpreting angles and rotations in ${\mathbb{R}}^{*}\times{\mathbb{R}}^{*}$, one can apply these ∗maps. Yet, that way of introducing trigonometry is hardly popular among geometry scholars.

While dealing with angles in a Cartesian plane over a Pythagorean field, i.e., closed under the operation $x\mapsto\sqrt{1+x^{2}}$, Hilbert’s solution was the following [8, §9]. Let points $O,E,C$ have coordinates $(0,0)$, $(1,0)$, $(c_{x},c_{y})$. Then the rotation of the angle $\angle EOC$ about $O$ turns point $(x,y)$ into $(x^{\prime},y^{\prime})$ defined as follows

[TABLE]

[TABLE]

Thus, terms $\frac{c_{x}}{\sqrt{c_{x}^{2}+c_{y}^{2}}}x$ and $\frac{c_{y}}{\sqrt{c_{x}^{2}+c_{y}^{2}}}y$ are to mimic the cosine and sine of the angle $\angle COE$, nonetheless, they make sense in any Pythagorean field.

Considering angles and rotations in a Cartesian plane, Hartshorne mimics the real function $tan$.555See [7, §16–17]. Greenberg excludes the Poincare length in a non-Archimedean, Euclidean field.666See [6, 320]. Max Dehn introduced semi-Euclidean planes over non-Archimedean Pythagorean fields. And indeed, for a long time, non-Archimedean, Euclidean fields have not been known, and it could be a reason why Hilbert did not introduce the circle-circle axiom into his system of axioms. The only non-Archimedean fields known at that time were Pythagorean fields.

4. Limiting parallel rays

In this section we show that the plane $\mathbb{L}\times\mathbb{L}$ is not hyperbolic by considering limiting parallel rays.

Take the point $A=(0,\varepsilon)$, for some positive infinitesimal $\varepsilon$. Rays $y_{1}=\varepsilon^{2}x+\varepsilon$ and $y_{2}=-\varepsilon^{2}x+\varepsilon$ contain $A$ and do not meet $y_{3}\equiv 0$. Suppose $y=\delta x+\varepsilon$ is the limiting ray, that is, such that every line inside the angle with arms $y$ and $AO$ meets $y_{3}$. Slope $\delta$ has to satisfy the condition $\tfrac{-\varepsilon}{\delta}\in\Psi$. However, $y_{5}=\sqrt{\delta}x+\varepsilon$ lies inside the angle with arms $y$ and $AO$, contains $A$ and does not meet $y_{3}$. Therefore, $\mathbb{L}\times\mathbb{L}$ is not a hyperbolic plane.

Usually, non-Euclidean and not hyperbolic planes are modeled through some tricks concerning the algebraic properties of a field. It could be, for example, a Poincaré disc over a Pythagorean field that is not Euclidean. Our plane $\mathbb{L}\times\mathbb{L}$ is not hyperbolic because it is semi-Euclidean. Nevertheless, one can also observe that both Euclidean and hyperbolic parallel axioms fail because straight lines in that plane are too ‘short’.