Looking for Geometry

In the Wonderful World of Dance

“A dignified formal dance is delicately planned Geometry”

-Ruth Katz

Dance as an Interdisciplinary Toola form of learningfacilitates developmentAlternative Integrates

Geometry and Dance

Elements of Geometry are used as

Elements Choreography

Choreography

Attention is paid to the form, look, shape, and feelManipulating time, energy, and space

Manipulation of Space

Space design and dance structure evolve together through the use of space elementsThese elements are shape/line, level, direction, focus, points on stage, floor patterns, depth/width, phrases and transitions

Shape and Line

Shape of DanceShape of Movements Choreography Decides

Shapes and Lines

Curved

Angular

Linear

Symmetrical and Asymmetrical

In symmetrical design the body parts are equally proportioned in space In asymmetrical designs the body parts are not equally proportioned in space

Points on Stage

Upstage RightUpstage CenterUpstage LeftCenter RightCenter Center LeftDownstage RightDownstage CenterDownstage Left

George Balanchine

1904- 1983

George Balanchine

Founder of the School of American Ballet (1934) and the New York City Ballet (1948)Major figure in Mid-20th Century BalletCreated 425 dance works Classical Modern American Style Ballet Dances free from symmetrical form Celebrated for imagination and originality

George Balanchine (cont.)

Shifting geometric patterns Stressed straight lines Serenade (1934)The Nutcracker (1954)Symphony in Three Movements (1972)Stravinsky’s Variation for Orchestra (1982)

Using dance as a Reinforcement

Primary Grades-dance can be used to reinforce diagonals,

vertical and horizontal lines on a plane

Secondary Grades-recreate the concepts of symmetry and asymmetry, shown on paper, and recreate

them through movement and dance“Rhythm and Symmetry are the connectors between Dance and Math”

-Dance can reinforce geometry’s basic concepts and construction.

• Primary and Secondary

Geometry in Art

By:Laura Szymanik

DefinitionsPolygon: Union of segments in a plane meeting only at endpoints or the vertex points.

Polyhedron: is an object that has many faces, also known as platonic solids.

Types of Regular Polygons

pentagon (5 sided) hexagon (6 sided) heptagon (7 sided) octagon (8 sided) nonagon (9 sided) decagon (10 sided)

Polygons and Pyramids

Polygons and pyramids are another example of polyons. A pyramid has two bases and rectangular faces to close it. A pyramid has one base and triangular faces to close it. The faces meet at one point called the apex.

Types of Regular Polyhedrons

cube (face is a cube)

tetrahedron (face is an equilateral triangle)

octahedron (face is an equilateral triangle)

icosahedron (face is an equilateral triangle)

dodecahedron (face is a regular pentagon)

                                                              

CUBE

OCTAHEDRON DODECAHEDRON

TETRAHEDRONICOSAHEDRON

Relationship Between Polygons and Polyhedrons

A polyhedron and polygon share some of the same qualitites. A regular polyhedrons face is the shape of a regular polygon.

For example: A tetrahedron has a face that is an equilateral triangle. This means that every face that makes the tetrahedron is an equilateral triangle. Around all the vertices and every edge is the same equilateral triangle.

Relationship Between Polygons and Polyhedrons

A polyhedron is made of a net which is basically like a layout plan. It is flat and made of all the faces that you will see on the polyhadron.

For example: A cube has six faces all of them are squares. When you open the cube up and lay it out flat you see all the six squares that it is made of.

Examples in Art

Leonardo da Vinci’s Polyhedras

Tessellations in Art

Ginger Baker

What is a Tessellation?

Definition

A dictionary* will tell you that the word "tessellate" means to form or arrange small squares in a checkered or mosaic pattern. The word "tessellate" is derived from the Ionic version of the Greek word "tesseres," which in English means "four." The first tilings were made from square tiles.

Different types of Tessellations

A regular tessellation means a tessellation made up of congruent regular polygons. [Remember: Regular means that the sides of the polygon are all the same length. Congruent means that the polygons that you put together are all the same size and shape.]

Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons.

Continued...

Semi-regular Tessellations

You can also use a variety of regular polygons to make semi-regular tessellations. A semiregular tessellation has two properties which are:

It is formed by regular polygons. The arrangement of polygons at every

vertex point is identical.

M.C. Escher

Popular artist who used tesselations oftenThe twists and turns of the human mind were broughtto life in the work of Dutch artist M.C. Escher.

His artwork is a mix of distorted perspectivesand optical illusions.

Impossible angles, connections, and shapes were Escher's favorite subjects.

M.C. Escher

M.C. Escher was born in the European country ofThe Netherlands on June 17, 1898.

His art is famous all over the world.

Although he was not trained as a mathematician or scientist, you may have seen one of Escher's works on the wall of your math class at school.

His work was respected by both mathematicians and artists.

M.C. Escher

His use of patterns, images that change into one another and perspectives are fascinating.

Escher also created tesselations, or interlocking patterns.

Some of Escher’s Work

Day and Night

Sky and Water

76 Horse Symmetry

Other artists

If you are fascinated by the work of the late Dutch artist M. C. Escher, the recognized master of the two dimensional planar tessellation or regular division of the plane, then you should also enjoy Seattle graphic artist K. E. Landry's work.

Landry's inventory of 2D tessellation images include both natural creatures and geometric art demonstrate congruent objects arranged with symmetry that often challenge the eyes ability to pick out the "members of the cast."

Landry

Landry has gone "Beyond Escher" to design his tessellation images with internal geometry that allows a dissection of the planar components, which after folding and pasting, allows an onlay, inlay, or overlay of many of the convex 3D spatial shapes including the, Platonic, Prism and Antiprism, and Archimedian polyhedra.

These 'enhanced' polyhedra are called the "Decorated Polyhedra."

Landry

Landry

More Landry

Resources

www.mathforum.orgwww.johnshepler.com/posters/escherpicturewww.landryart.com

What is the golden section (or Phi)? We will call the Golden Ratio (or Golden number) after a greek letter,Phi although some writers and mathematicians use another Greek letter, tau. Also, we shall use phi (note the lower case p) for a closely related value. A simple definition of Phi There are just two numbers that remain the same when they are squared namely 0 and 1. Other numbers get bigger and some get smaller when we square them: Squares that are biggerSquares that are smaller22 is 4 1/2=0·5 and 0·52 is 0·25=1/4 32 is 9 1/5=0·2 and 0·22 is 0·04=1/25 102 is 100 1/10=0·1 and 0·12 is 0·01=1/100 One definition of Phi (the golden section number) is that to square it you just add 1 or, in mathematics: Phi2 = Phi + 1

The Golden Section and Art

Luca Pacioli (1445-1517) in his Divina proportione (On Divine Proportion) wrote about the golden section also called the golden mean or the divine proportion:

A M B The line AB is divided at point M so that the ratio of the two parts, the smaller MB to the larger AM is the same as the ratio of the larger part AM to the whole AB.

Phi and the Golden Section in Art

As the golden section is found in the design and beauty of nature, it can also be used to achieve beauty and balance in the design of art.  This is only a tool though, and not a rule, for composition.The golden section was used extensively by Leonardo Da Vinci.  Note how all the key dimensions of the room and the table in Da Vinci's "the last supper" were based on the golden section

The French impressionist painter Georges Pierre Seurat is said to have "attacked every canvas by the golden section

Note that successive divisions of each section of the painting by the golden section define the key elements of composition. 

The horizon falls exactly at the golden section of the height of the painting.  The trees and people are placed at golden sections of smaller sections of the painting. 

The Golden Section in Art

The Golden Section in Art

Golden Section