Images of Escher Drawings Hyperbolic Plane

Contents

• 1 Explorations
• 2 Hyperbolic Infinite
• 3 Models of Hyperbolic Space
  • three.i Poincare Deejay Model
  • three.2 Upper Half Space Model
  • iii.iii Other Models
• 4 Polygons and Defect
  • 4.1 Ideal Polygons
• 5 Hyperbolic Tessellations
  • 5.i Examples
  • five.2 Escher's Circle Limit Iii
  • 5.iii Platonic Tessellations
• vi Exercises
• 7 Relevant examples from Escher's work
• 8 Related Sites
• 9 Notes

Explorations

You may begin exploring hyperbolic geometry with the following explorations. We recommend doing some or all of the basic explorations earlier reading the section. Gaining some intuition about the nature of hyperbolic infinite before reading this department will be more effective in the long run.

Basic Explorations

• Hyperbolic Paper Exploration
• Escher's Circle Limit Exploration This exploration is designed to help the student gain an intuitive understanding of what hyperbolic geometry may look like. Escher's prints are nice examples that illustrate what we would encounter when looking downwardly on a hyperbolic universe.
• Hyperbolic Geometry Exploration NonEuclid allows united states of america to depict acurate pictures of objects in hyperbolic space.

Farther explorations that assist requite a amend agreement of hyperbolic geometry.

• Hyperbolic Geometry II with NonEuclid Exploration
• Hyperbolic Tessellations Exploration
• Ideal Hyperbolic Tessellations Exploration
• Hyperbolic Escher Exploration

Hyperbolic Space

We take seen two different geometries so far: Euclidean and spherical geometry. Geometry is meant to describe the earth effectually us, and the geometry then depends on some central properties of the world we are describing. Objects that live in a flat world are described past Euclidean (or apartment) geometry, while objects that live on a spherical world will demand to be described by spherical geometry.

In 2 dimensions there is a tertiary geometry. This geometry is called hyperbolic geometry. If Euclidean geometry describes objects in a flat world or a aeroplane, and spherical geometry describes objects on the sphere, what world does hyperbolic geometry describe? Similar spherical geometry, which takes place on a sphere, hyperbolic geometry takes place on a curved two dimensional surface called hyperbolic space.

We will describe hyperbolic space in several different ways. In Escher's work, hyperbolic space is a distorted deejay. All of the angels in Circle Limit Iv (Heaven and Hell) alive in hyperbolic space, where they are really the same size, as do the devil figures. The image that Escher presents is a distorted map of the hyperbolic earth.

You tin can explore Escher's hyperbolic Circle Limit prints and get an introduction to hyperbolic geometry in the Escher's Circle Limit Exploration

Models of Hyperbolic Space

On a sphere, a pocket-size neighborhood of a point looks like a cap. In hyperbolic space, every point looks similar a saddle:

A piece of a sphere	A piece of hyperbolic space

Unfortunately, while you can slice caps together to brand a sphere, piecing saddles together chop-chop runs out of infinite. Attempt this yourself with the Hyperbolic Newspaper Exploration.

Hyperbolic newspaper is a floppy, saddle like object. Eventually, it contains as well many triangles in besides small a infinite to continue any further, although most people and run out of patience before running out of room. It'south besides possible to crochet models of hyperbolic space.

Because globes are unwieldy, navigators employ flat maps of the spherical world. Maps of the Earth are necessarily distorted, for example Greenland appears extremely large on the common Mercator map of the Earth, shown with cherry-red dots to point the distortion of expanse.

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Mercator projection map with equal area dots.

Poincare Disk Model

Because models of hyperbolic infinite are unwieldy (not to mention infinite), we will do all of our piece of work with a map of hyperbolic space chosen the Poincaré disk. The Poincaré disk is the inside of a circumvolve (although the circle is not included) and is badly distorted near its edge.

Objects nearly the edge of the Poincaré deejay are larger than they appear.

Hyperbolic man takes a walk

The picture shows a stick man as he walks towards the edge of the disk. He appears to compress, as does the altitude he moves with each step. Merely this deejay is a distorted map, and in the actual hyperbolic infinite his steps are all the aforementioned length and he stays the same size. The man will never reach the edge of the deejay, because it is infinitely far away. The border is drawn dashed because information technology is not actually part of hyperbolic space.

The geodesics in hyperbolic space play the role of straight lines. Geodesics announced straight to an inhabitant of hyperbolic space, and they are the shortest paths between points. In the Poincaré disk model, geodesics announced curved. They are arcs of circles. Specifically:

Geodesics are arcs of circles which meet the edge of the disk at 90°.

Geodesics which pass through the middle of the disk appear straight.

Some geodesics in the Poincaré disk

Exercise drawing geodesics in the Poincaré disk with Hyperbolic Geometry Exploration.

Upper Half Space Model

Ideal hyperbolic triangle in the upper one-half space model. From a Roman mosaic, Real Alcazar, Cordoba, Spain

Another commonly used model for hyperbolic infinite in the upper half infinite model. In this model, hyperbolic space is mapped to the upper half of the plane. The model includes all points (ten,y) where y>0. In other words, everything above the x-axis.

The geodesics in the upper half space model are lines perpendicular to the ten-axis and semi-circles perpendicular to the x-axis. The prototype of the mosaic to the right shows iii geodesics. The lighter semi-circles at the bottom create 2 geodesics, and the dark semi-circle in the background creates the 3rd geodesic. Geodesics measure shortest distance and play the role of straight lines in Euclidean geometry, hence these three geodesics form an (platonic) triangle. Notation how the geometric figures have rather small bending measures just as in the Poincaré deejay model discussed above.

Non-Euclid also allows the user to experiment with this model of hyperbolic space.

Other Models

In that location are several other models that can be used to represent hyperbolic infinite. The Pseudosphere is a model that accurately shows how hyperbolic space curves, but but models a portion of the whole space.

The Beltrami-Klein model represents hyperbolic infinite as the interior of a disk, just equally the Poincare disk model, but it chooses to misconstrue angles rather than geodesics. So the geodesics actually appear equally straight lines, merely the angles between them are no longer correctly shown.

Enough of other models exist, just like there are many means to make maps of the spherical geometry of the Earth, just the Poincare disk model is the ane that Escher uses exclusively.

Polygons and Defect

Polygon

A polygon in hyperbolic geometry is a sequence of points and geodesic segments joining those points. The geodesic segments are called the sides of the polygon.

A triangle in hyperbolic geometry is a polygon with three sides, a quadrilateral is a polygon with four sides, and then on, equally in Euclidean geometry. Here are some triangles in hyperbolic space:

From these pictures, you can see that:

The sum of the angles in whatsoever hyperbolic triangle is less than 180°.

Defect

The defect of a hyperbolic triangle is 180° – (angle sum of the triangle).

By cut other polygons into triangles, we see that a hyperbolic polygon has angle sum less than that of the corresponding Euclidean polygon.

Define the defect for a hyperbolic polygon with <math>n</math> sides to be <math>(northward - 2)\times180^\circ - (\text{angle sum of the polygon})</math>.

Putting this together with the defect in spherical geometry:

The defect of a polygon is the difference between its angle sum and the angle sum for a Euclidean polygon with the same number of sides.

This statement works in spherical and hyperbolic geometry, for polygons with any number of sides. It even works for biangles, considering a biangle in Euclidean geometry must take two 0° angles. The surface area of a hyperbolic polygon is notwithstanding proportional to its defect:

Surface area of a hyperbolic polygon = <math>\frac{\pi}{180^\circ}\times \text{defect}</math>.

This equality is a special case of the Gauss-Bonnet theorem.

In spherical geometry, we had a formula relating the defect of a polygon to the fraction of the sphere's surface area covered past the polygon. For a sphere of radius i, the total surface expanse of the sphere is <math>4\pi</math>, and and then the surface area of a polygon is <math>4\pi \times \frac{\text{defect}}{720^\circ}</math> which (after a little simplification) is exactly the same formula as in hyperbolic infinite!

Ideal Polygons

Ideal Triangle

An ideal triangle consists of three geodesics that touch at the boundary of the Poincaré deejay.

The three geodesics are called the sides of the platonic triangle. Since the boundary of the disk isn't role of hyperbolic space, the sides of an ideal triangle are infintely long and never really meet. Still, the do get closer together as they head towards the edge. The iii points on the boundary are called the platonic vertices of the platonic triangle, and play a similar role every bit the vertices of an ordinary triangle. Since the sides are all perpendicular to the Poincaré disk boundary, they make an bending of 0° with each other.

You can create other ideal polygons in a similar manner:

Ideal triangles and an ideal hexagon

The area formula implies that whatsoever ideal triangle has area <math>\pi</math>, because the bending sum is zero and its defect is 180°. Both ideal triangles shown above accept the same area even though the distortion of the Poincaré disk makes one await much smaller than the other. The platonic hexagon shown has angle sum zippo, so it's defect is 720° and its surface area is <math>4\pi</math>.

Hyperbolic Tessellations

Hyperbolic Tessellations Exploration

A hyperbolic tessellation is a covering of hyperbolic space by tiles, with no overlapping tiles and no gaps. Escher'south Circumvolve Limit prints are examples of hyperbolic tessellations. Like his other tessellations, Escher began with a geometric tessellation by polygons and worked from at that place.

Consider Escher'south Circumvolve Limit I, shown with geometric scaffolding consisting of geodesics drawn in red:

The spines of the fish, emphasized with red lines, class a tessellation of hyperbolic infinite past quadrilaterals. Since these quadrilaterals run across four or six at a vertex, they have corner angles 90°-sixty°-90°-lx°. They are not regular polygons because regular polygons have all sides and all angles equal.

There are plenty of other tessellations of hyperbolic infinite, including regular tessellations. In fact, there are infinitely many regular tessellations of hyperbolic space. This stands in abrupt contrast to Euclidean space, which has only three, and to spherical geometry, where there are simply 5 not-degenerate possibilities (corresponding to the Ideal solids). Hyperbolic infinite is like shooting fish in a barrel to tessellate because the corner angles of polygons desire to be small, and small angles fit nicely around a vertex.

As in the other two geometries, we describe regular tessellations past the number of sides in each polygon and the number of polygons that come across at a vertex.

Schläfli Symbol

The Schläfli symbol <math>\{n,m\}</math> describes the regular tessellation by <math>n</math>-gons where <math>k</math> meet at each vertex.

Examples

This tessellation has seven triangles at each vertex, and so its Schläfli symbol is {3,7}.

The triangles have

360°/vii ≈51.4°

angles and angle sums of approximately

3×51.4°=154.3°

This gives a defect of approximately 180°–154.3° = 25.7° and an surface area of approximately 0.45.

This tessellation has 4 pentagons at each vertex, then its Schläfli symbol is {5,four}.

Each pentagon has five 90° angles.

The angle sum for each pentagon is 450°.

Since Euclidean pentagons accept bending sum 540°, these pentagons take a defect of 540°-450° = ninety° and an area of <math>\pi/two \approx ane.57</math>.

This tessellation has six quadrilaterals at each vertex, and and then its Schläfli symbol is {four,half-dozen}.

Each quadrilateral has four 60° angles.

The bending sum for each quadrilateral is 240°.

Since Euclidean quadrilaterals have angle sum 360°, these quadrilaterals have a defect of 360°-240° = 120° and an area of <math>ii\pi/iii \approx 2.09</math>.

Escher'southward Circle Limit 3

Escher's Circle Limit III is his nearly achieved print based on hyperbolic space. It is also his well-nigh subtle.

Looking at the white spines of the fish, it appears to exist a tessellation of hyperbolic space past triangles and squares, with iii triangles and three squares coming together at each vertex. This would mean that the corner angles of both polygons are 60° each, but this gives the triangle an angle sum of 180°, which cannot happen in hyperbolic space!

Still, Circle Limit Three clearly has the season of a hyperbolic tessellation. H.S.M Coxeter, a mathematician and friend of Escher's analyzed the print^[1], and discovered that the white circular arcs forth the spines encounter the boundary of the disk at eighty° each, rather than the 90° required to be geodesic. Since the white lines are not geodesics, neither the "triangle" or the "foursquare" is actually a polygon at all! Even Escher does not seem to have realized this, although almost probably he would not accept cared, every bit he was very satisfied with the impress and the proposition of infinity it presents.

A natural underlying tessellation for Circle Limit III is the {viii,3} tessellation by octagons:

A geodesic and two (non-geodesic) curves at equal distance from the geodesic.

The image on the right shows a hyperbolic geodesic that runs through the midpoints of the sides of the octagons, surrounded by ii curves (not geodesics) at a stock-still distance from the geodesic. The rightmost of these curves forms the spines of one row of fish in Circle Limit Three.

Ideal Tessellations

A hyperbolic tessellation by ideal triangles.

Cartoon hyperbolic tessellations by hand is quite difficult. However, tessellations by ideal polygons are somewhat easier to work with. The advantage of an ideal tessellation is that all the vertices lie on the boundary (or, more precisely, approach the boundary) of the Poincaré disk. Since the tiles in ideal tessellations are not polygons, but ideal polygons, they do not accept Schläfli symbols. However, they do fit into the scheme of regular tessellations, with the odd feature that infinitely many tiles now run into at an platonic vertex.

Acquire more most these in the Ideal Hyperbolic Tessellations Exploration.

Exercises

Hyperbolic Geometry Exercises

Relevant examples from Escher's work

• Circle Limit I
• Circle Limit Ii
• Circle Limit III
• Circle Limit Iv (Heaven and Hell)

• NonEuclid: A coffee applet for working with the Poincaré disk, by Joel Castellanos.
• Hyperbolic Geometry Applet: by Paul Garrett
• HyperRogue: An action/strategy game that takes place in the Poincaré disk, by Zeno Rogue Games.
• Online Showroom: Hyperbolic Space by The Institute For Figuring.
• CurvedSpaces A non-Euclidean space flight simulator by Jeff Weeks.
• Doug Dunham has a collection of articles on hyperbolic tessellations and Escher's Circle Limit series.
• Hyperbolic Tessellations, by David Joyce.
• Hyperbolic Tessellations Applet by Don Hatch.
• Capturing Infinity, the Circle Limit Serial of M.C. Escher, by T. Wieting, Reed Mag 2010.

Notes

1. ↑ Coxeter, H.S.Thou. "The Non-Euclidean Symmetry of Escher's Picture 'Circle Limit III'". Leonardo Vol. 12 No. one, 1979, pg 19-25.
2. ↑ Dunham, D. "More 'Circle Limit III' patterns". Proceedings of Bridges 2006 pg 451.

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Source: https://mathstat.slu.edu/escher/index.php/Hyperbolic_Geometry

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