William Thurston designed a paper model of the hyperbolic plane. However his models were very fragile and difficult to make.

In 1997, Dr. Diana Tiamina's models made it possible to actually feel hyperbolic space and see many of its properties in action by using crochet.

Hyperbolic Soccer ball

ACTIVITY:

Models of Hyperbolic Plane

**APPLICATIONS ON HYPERBOLIC PARABOLOID**

Tools in Domestic Use

• All triangles have angle sum less than 180.

• Rectangles do not exist.

• If two triangles are similar they are congruent.

• All convex quadrilaterals have angle sum less than 360.

• The hyperbolic axiom holds for all lines.

The Poincaré disk is a model for

hyperbolic geometry in which a line is

represented as an arc of a circle whose

ends are perpendicular to the disk's

boundary (and diameters are also

permitted).

Arts

What are some important properties of hyperbolic geometry?

The upper half plane model takes the Euclidean upper half plane as the "plane." The "lines" are portions of circles with their center on the boundary.

There is an isomorphism between the Poincaré hyperbolic disk model and the Klein-Beltrami model.

The Klein-Beltrami model of hyperbolic

geometry consists of an open disk in

the Euclidean plane whose open

chords correspond to hyperbolic lines.

To visualize hyperbolic geometry and its properties we ought to discard the flat plane image of the Euclidean plane and think in terms of a shape like that of the saddle.

Physical Realizations in Nature

The hyperbolic axiom states that there exists a line 1 and an external point not lying on this line such that there are at least 2 lines passing through this point and parallel to this line 1.

Physical Realizations in Nature

Food Industry

Hyperbolic plane geometry is the geometry of saddle

surfaces with constant negative Gaussian curvature

(for example the pseudosphere).

INTRODUCTION

BASIC

CONCEPTS

APPLICATION

Science and

Architecture

THE HYPERBOLIC AXIOM

What is Hyperbolic Geometry?

It is obtained by

considering only the first four postulates of Euclid and

replacing the parallel postulate by its negation, called the hyperbolic axiom.

1

Poincare Model

2

Upper Half Plane Model

3

Klein-Beltrami Model

The Dutch artist M. C. Escher is known for his repeating patterns of interlocking motifs, tessellations of the Euclidean and the hyperbolic plane and his drawing representing impossible figures.

The

Poincaré Ball

is the 3D analogy of the Poincaré disk. It is a projection of uniformly tiled hyperbolic polyhedrons from hyperbolic space into Euclidean space.

Poincaré Hyperbolic Tiling contains tiles characterized as 7-sided polygon, and 3 polygons. These meet at each vertex and are generated using a series of reflections called anti-homographs

Note: The hyperbolic paraboloid (not to be confused with a hyperboloid) is a doubly ruled surface shaped like a saddle.

“Why are Pringles a hyperbolic paraboloid?”

to be stacked in sturdy tubular containers

for easier stacking of chips.

minimizes the possibility of broken chips during transport.

increases the crunchy feeling.

relatively more feasible to manufacture the press block compared to other shapes.

strikes a delicate balance between these push and pull forces, allowing it to remain thin yet surprisingly strong

Architecture

**Science**

Mercator projection

Gravitational Lensing

Cosmology

Theory of General Relativity

**By Group 17**

Math 1- Ma'am walo

Singson

sinoy

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soriano

SURQUIa

tagud

Math 1- Ma'am walo

Singson

sinoy

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soriano

SURQUIa

tagud

References

http://i2.cdn.turner.com/cnn/2010/images/04/29/c1main.holly.jpg

http://www.ux1.eiu.edu/~cfcid/Classes/3271/Presentations/lettucecoral.jpg

http://i2.cdn.turner.com/cnn/2010/images/04/29/c1main.zj7.jpg

http://www.homeschoolmath.net/blog/Glossodoris_atromarginata.jpg

https://darlombfiberart.files.wordpress.com/2013/02/daina-taimina-crocheted-hyperbolic-plane.jpg

https://s-media-cache-ak0.pinimg.com/736x/a6/6e/67/a66e67493317a7a7d730171ec38f1272.jpg

http://01sj.org/files/2010/03/Nov6-09.jpg

http://uh8yh30l48rpize52xh0q1o6i.wpengine.netdna-cdn.com/wp-content/uploads/2013/03/49923_8dbffc89b3188752_b.jpg

http://www.math.cornell.edu/~mec/Winter2009/Mihai/section5_files/circle_limit_1.jpg

http://www.math.cornell.edu/~mec/Winter2009/Mihai/section5_files/circle_limit_2.jpg

http://www.math.cornell.edu/~mec/Winter2009/Mihai/section5_files/circle_limit_3.jpg

http://www.math.cornell.edu/~mec/Winter2009/Mihai/section5_files/circle_limit_4.jpg

http://bugman123.com/Hyperbolic/PoincareBall2-large.jpg

http://bugman123.com/Hyperbolic/PoincareBall.jpg

http://bugman123.com/Hyperbolic/PoincareSurfaceKnot7-3-large.jpg

http://stradasaddles.co.uk/images/st-christopher-large.png

http://stradasaddles.co.uk/images/st-christopher-large.png

https://motorsportsnewswire.files.wordpress.com/2014/02/140213-corbin-bmw-f8gs-14-adv_2.jpg

http://www.sciencemuseum.org.uk/HoMImages/Components/1093/109346_3.png

https://www.google.com.ph/imgres?imgurl=http%3A%2F%2Fresearch.microsoft.com%2Fen-us%2Fum%2Fpeople%2Fyangliu%2Fgeometry%2Fimages%2Fdesign.JPG&imgrefurl=http%3A%2F%2Fresearch.microsoft.com%2Fen-us%2Fum%2Fpeople%2Fyangliu%2Fgeometry%2F&docid=wSSYhumChCk1CM&tbnid=lD5iHms6s8gv0M%3A&w=690&h=357&ved=0ahUKEwii06uioYnMAhVIo5QKHYl2BAwQMwggKAUwBQ&iact=mrc&uact=8

https://www.google.com.ph/imgres?imgurl=http%3A%2F%2Fassets.inhabitat.com%2Fwp-content%2Fblogs.dir%2F1%2Ffiles%2F2012%2F03%2FBarkow-Leibinger-Local-Techniques-Geometrical-Installation-Marrakech-6-537x358.jpg&imgrefurl=http%3A%2F%2Finhabitat.com%2Fbarkow-leibinger-unveils-striking-loom-hyperbolic-art-installation-made-from-local-natural-materials%2F&docid=3_M37YoEgYHimM&tbnid=un3t8K6GWXCheM%3A&w=537&h=358&ved=0ahUKEwii06uioYnMAhVIo5QKHYl2BAwQMwgdKAIwAg&iact=mrc&uact=8

https://www.google.com.ph/imgres?imgurl=https%3A%2F%2Fc2.staticflickr.com%2F6%2F5335%2F9763167215_0a7aaf27be_b.jpg&imgrefurl=https%3A%2F%2Fwww.flickr.com%2Fphotos%2Fhobbiepics%2F9763167215&docid=WL7Sw2EtNZrfyM&tbnid=O2VWL-GzE-yeFM%3A&w=1024&h=577&itg=1&ved=0ahUKEwii06uioYnMAhVIo5QKHYl2BAwQMwg5KBQwFA&iact=mrc&uact=8

http://euler.slu.edu/escher/index.php/File:Tissot_mercator.png

https://www.google.com.ph/imgres?imgurl=http%3A%2F%2Fi.huffpost.com%2Fgen%2F1117766%2Fthumbs%2Fo-GRAVITY-WAVES-570.jpg%3F6&imgrefurl=http%3A%2F%2Fwww.huffingtonpost.com%2F2013%2F05%2F03%2Fgravity-waves-found-einstein-general-relativity-

http://www.roe.ac.uk/~heymans/website_images/Gravitational-lensing-galaxyApril12_2010-1024x768.jpg

A TRUNCATED ORDER-7 TRIANGULAR TILING

Note:

In hyperbolic geometry, a uniform (regular, quasiregular or semiregular) hyperbolic tiling is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive.

This is only possible using tilings with 3 polygons meeting at each vertex because bubble intersections

must always meet at exactly 120° angles.