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Polyhedra “There Is Geometry in the Humming of Strings, There Is Music in the Spacing of the Spheres.” - Pythagoras

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Polyhedra “There Is Geometry in the Humming of Strings, There Is Music in the Spacing of the Spheres.” - Pythagoras Polyhedra “There is geometry in the humming of strings, there is music in the spacing of the spheres.” - Pythagoras Math 110 Math Applications Art, Fall 2018 A Quick Review of Polygons Edge • A closed, connected chain of (straight) line segments. Face The line segments are called edges or sides • Vertex • The points where two segments meet are called vertices, nodes, or corners Interior Angle • The area surrounded by the segments is called the face • The path created by the sides is called the Below are NOT polygons: boundary or perimeter • An interior angle is measured within the face, and is formed by two sides Sides not straight Not connected Not closed Math 110 Math Applications Art, Fall 2018 Classifying Polygons • Polygons are named for the number of sides they have: triangle/trigon (3), quadrilateral (4), pentagon (5), hexagon (6), etc. • An n-gon is a polygon with n sides • If all edges have the same length, the polygon is called equilateral • If all angles have the same measure, the polygon is called equiangular • If the polygon is BOTH equilateral and equiangular, it is called regular • NOTE: Regular is sometimes used to imply only equilateral. In these slides, assume regular means both “equal side length” and “equal angles” Math 110 Math Applications Art, Fall 2018 Classifying Polygons • A polygon is convex if it is NOT possible to draw a line segment that begins and ends within the polygon, but leaves the polygon at some point in between Convex: Not convex: Any line segment we draw will We can find at least one line segment that stay entirely inside the shape begins and ends in the shape, but leaves in between Math 110 Math Applications Art, Fall 2018 Tiling • A tiling is a “plane filling” arrangement of shapes, so that the shapes do not overlap and cover the surface entirely • The edges of each shape should be entirely against another edge, so “brick-laying” patterns do not count • Only three regular (equilateral, equiangular, convex) polygons can tile the plane: the square, the equilateral triangle, and the regular hexagon: Math 110 Math Applications Art, Fall 2018 Tiling: Why Only Three? • The vertices of the shapes have to meet together so that the sum of the interior angles is 360 • Less than that, and there will be a gap • More than that, and the shapes will overlap We need this to be a whole number • So the shape only works if 360 is a multiple of its interior angle Sides, n Angle, Θ 360/Θ Tiles? Interior Angle Formula: 3 60 6 Yes 180(n 2) 4 90 4 Yes ✓ = − 5 108 3.33 No n 6 120 3 Yes 7 128.57 2.8 No 8 135 2.66 No 9 140 2.57 No 10 144 2.5 No Math 110 Math Applications Art, Fall 2018 Relaxing Requirements • There are many more tilings we can create if we ignore one or more of the requirements • We’ll spend more time on these later - they are much more interesting! A pentagonal tiling, just not regular Mixing polygons - but al are regular Edges do not line up, but Non convex, but regular. The edges and convex, and the edges line up otherwise ok do not line up, either. Math 110 Math Applications Art, Fall 2018 Polyhedron (plural: Polyhedra) • A three dimensional solid figure with polygonal faces joined together by straight edges • Categorized in similar ways to the polygons: • Number of faces? • Types of faces: Are they all the same shape? Same size? • Convex? Math 110 Math Applications Art, Fall 2018 Polyhedron Anatomy • Face: one of the polygons that makes up the surface of the shape • Vertices: point where the vertices of the polygons meet together • Edge: line segment where edges of the polygons meet together Face Vertices Edge Math 110 Math Applications Art, Fall 2018 Polyhedron Naming • Naming is similar to polygons, but refers to the number and types of faces • More complicated naming schemes exist that reflect some of the other properties of the polyhedra, such as convexity • Some are given “nicknames,” like the cube Great Dirhombicosidodecahedron, Hexahedron, “Cube” “Miller’s Monster” Math 110 Math Applications Art, Fall 2018 Polyhedron Classification • A polyhedron is called truncated if it is created by chopping off each vertex, leaving additional faces Tetrahedron Truncated Tetrahedron Math 110 Math Applications Art, Fall 2018 Polyhedron Classification • A polyhedron is called stellated if it is created by “pulling” the center point of each face, forming a star-like shape Dodecahedron Stellated Dodecahedron Math 110 Math Applications Art, Fall 2018 Polyhedron Classification • A polyhedron is called regular if all of its faces are identical (congruent) regular, convex polygons This dodecahedron is regular. This icosidodecahedron is not regular. All of its faces are identical pentagons. Its faces are convex regular polygons, but they are not all identical. Math 110 Math Applications Art, Fall 2018 Polyhedron Classification • A polyhedron is called convex if it is NOT possible to draw a straight line segment beginning and ending in the shape that leaves the shape at any point This dodecahedron is convex This stellated dodecahedron is not convex Math 110 Math Applications Art, Fall 2018 Regular Convex Polyhedra • To build one, we’ll need to use regular convex polygons • We’ll have several convex polygons meeting together at each vertex • The sum of their interior angles must be less than 360 • If equal to 360, they’ll form a flat shape • If over 360, the shape curves the “wrong way” • There must be at least three polygons at each vertex, otherwise it’s flat Math 110 Math Applications Art, Fall 2018 Regular Convex Polyhedra • With equilateral triangles (interior angle 60), we have three options: • Three together (adds up to 180 degrees) • Four together (adds up to 240) • Five together (adds up to 300) • With squares (angle 90), we have only one option: • Three together (adds up to 270) • With pentagons (angle 108), we have one option: • Three together (adds up to 324) Math 110 Math Applications Art, Fall 2018 Regular Convex Polyhedra • Nothing else works! Sides, n Angle, Θ • 2 hexagons is not enough, 3 is too many 3 60 4 90 • Same for any other polygon with more than 5 108 5 sides - three together will always add up 6 120 to more than 360 degrees 7 128.57 8 135 9 140 10 144 Math 110 Math Applications Art, Fall 2018 Regular Convex Polyhedra • Only five possible regular convex polyhedra • These were studied by many cultures, and thoroughly document by Euclid in The Elements • They were of particular interest to Plato, and are now called the Platonic solids in his honor • Stone models of the solids have been found in Scotland, dating to the neolithic period (1,000 years before Plato) • Here’s a link to some models of the Platonic solids Math 110 Math Applications Art, Fall 2018 The Platonic solids • Plato associated each solid with an element: • Tetrahedron = Fire • Cube = Earth • Octahedron = Air • Icosahedron = Water • Dodecahedron: “the god used it for arranging the constellations on the whole heaven” Johannes Kepler’s drawings of the Platonic solids from his “Mysterium Cosmographicum” Math 110 Math Applications Art, Fall 2018 The Platonic solids • Incidentally, Kepler also played with the idea of platonic solids in astronomy • He believed (for a time) that the planets orbited the sun in a pattern predicted by the Platonic solids, nested within each other, with the sun at the center • He later rejected this idea because it was not precise • It was also based on the idea that there are only 6 planets in our solar system, another issue Math 110 Math Applications Art, Fall 2018 Euler’s Formula • Euler noticed a pattern for convex polyhedra concerning the number of their faces, edges, and vertices • It applies to any convex polyhedra, but is easier to notice in the Platonic solids Polyhedron Vertices Edges Faces Tetrahedron 4 6 4 Cube 8 12 6 Octahedron 6 12 8 Dodecahedron 20 30 12 Icosahedron 12 30 20 Math 110 Math Applications Art, Fall 2018 Euler’s Formula • If F is the number of faces, V is the number of vertices, and E is the number of edges, then V - E + F = 2 Polyhedron Vertices Edges Faces Tetrahedron 4 6 4 Cube 8 12 6 Octahedron 6 12 8 Dodecahedron 20 30 12 Icosahedron 12 30 20 Math 110 Math Applications Art, Fall 2018 Archimedes of Syracuse • Greek, born around 287 BC and died around 212 BC • That’s Syracuse in Sicily, not upstate NY! • Considered one of the greatest mathematicians of all time • Derived and proved formulas involving circles using techniques now found in calculus, long before the field was developed • Also famous for finding method of calculating volumes of irregular objects via water displacement - “Eureka!” • Created parabolic mirrors that reflected light to burn enemy ships approaching Syracuse called the Archimedes heat ray Killed by Roman soldier during Second Punic War, • Domenico Fetti’s “Archimedes Thoughtful” after declining a meeting with General Marcellus 1620 because he was too busy working on a math problem Math 110 Math Applications Art, Fall 2018 Archimedean solids • Also convex polyhedra, but can be made of more than one type of polygon (all polygons of the same type must be identical) • Only 13, shown by Archimedes himself (who also discovered some) Math 110 Math Applications Art, Fall 2018 Essentially Two Dimensional Polyhedra • Some polyhedra can be formed by taking any polygon and stacking it into a tower • You may have created your own as “block letters” • We also call these shapes prisms; the glass rainbow producing prism is an example Math 110 Math Applications Art, Fall 2018 Classification Theorem • Every bounded solid object is either essentially two dimensional, or
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