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Platonic Geogami By Katie Snyder: [email protected] Debra Zibreg: [email protected] Michelle Dynarski: [email protected] Connie Hadley: [email protected] Focus: Applications of Platonic Solids Introduction: Building is fun! Platonic solids created from origami can be used in both the 6th grade and Geometry curricula. In the 6th grade curriculum, students need to learn probability and Ancient History, and both can be learned through the use of Platonic solids. These solids are also a fun and useful tool for students in Geometry to learn about Euler’s Formula, surface area, volume, and interior and exterior angles of polyhedra. NYS Math Standards: 6.S.9 List possible outcomes for compound events. 6.S.10 Determine the probability of dependent events. 6.S.11 Determine the number of possible outcomes for a compound event by using the fundamental counting principle and use this to determine the probabilities of events when the outcomes have equal probability. A.G.1 Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle. Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only). G.PS.2 Observe and explain patterns to formulate generalizations and conjectures. G.CM.2 Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams. G.CN.8 Develop an appreciation for the historical development of mathematics. G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts. G.G.36 Investigate, justify, and apply theorems about the sum of the measures of the interior and exterior angles of polygons. G.G.37 Investigate, justify, and apply theorems about each interior and exterior angle measure of regular polygons. Objectives: Following the conclusion of this lesson, students should be able to: understand and apply concepts of probability; use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes; identify and justify geometric relationships, formally and informally. Materials Needed: Origami paper, directions, writing utensil Classroom Applications: 1. History of Platonic solids. 2. Construct Platonic solids using origami. 3. Labeling Platonic solids. 4. Differentiate between prisms and Platonic solids. 5. Euler’s Formula for convex polyhedra. 6. Surface area and volume 7. Interior and exterior angles 8. Probability A Brief History of Platonic Solids Figure 1 Tetrahedron Figure 2 Cube or Hexahedron Figure 3 Octahedron An Etruscan dodecahedron, dated at 500 BCE, was found in Padova, Italy, and Neolithic stones have been found in Scotland that resembles Platonic solids believed to date around 2000 B.C. The Pythagoreans (6th century B.C.) are known to have considered the solids in their quest in simplifying the world around them to numbers. Plato, from whom the solids received their name, thought the platonic solids made up the universe. It is believed that he developed this idea from the Pythagoreans, but this has not been proven with any solid evidence. He documented in his book The Timaeus (355 B.C.E) a conjecture that the earth was composed of cubes, air was composed of octahedrons, water was composed of icosahedrons, and fire was made of tetrahedrons. He also presumed that dodecahedrons made up the cosmos. Some believe this is because it has 12 faces which correspond to the zodiac. Others think that the golden ratio found in the pentagon face of the dodecahedron is the reason Plato associated it with the cosmos. Around 200 BCE, in Book 13 of Euclid’s Elements, Euclid proved that there can only be five Platonic solids. In 1597 Johannes Kepler published The Cosmographic Mystery, stating that the distances of the planets’ orbits were related to the Platonic solids. In the year 1600, Kepler was invited by Tycho Brahe to become his assistant in Prague. Upon reviewing Tycho’s astronomical observations, Kepler concluded that the orbits were elliptical instead, a view that we know to be true today. Around 1750 Euler discovered that if a polyhedron is topologically a sphere, then the number of vertices plus the number of faces minus the number of edges equals two. In 1794, Legendre proved this, giving us more information Figure 4 Dodecahedron Figure 5 Icosahedron Figure 6 Etruscan Dodecahedron: Emmer, Michele, Ed. The Visual Mind: Art and Mathematics. Cambridge: MIT Press, 1993. p216 Figure 7 Kepler's Plantonic solid model of the solar system from Mysterium Cosmographicum (1596) about the properties of regular polyhedra. Differentiating between Platonic Solids and Prisms The Five Platonic Solids - What makes them so special? These are the only possible regular polyhedra. They are all regular solids. The same number of sides meet at the same angles at each vertex, and they have identical polygons that meet at the same angles at each edge. Each Platonic solid is also identified as a convex polyhedron because each face lies in a plane that does not intersect the rest of the polyhedron. Cube Tetrahedron # Vertices # Edges Octahedron # Faces Dodecahedron # of edges at each vertex n-gon for sides Icosahedron # of faces and face shape name Tetrahedron Cube Octahedron Dodecahedron Icosahedron Prisms: A prism is formed with two parallel congruent polygons (the bases - top and bottom) that are connected at the edges (the bases - top and bottom) with rectangles. Triangular Prism Rectangular Prism Pentagonal Prism # # Vertices Edges Triangular Prism Rectangular Prism Pentagonal Prism Hexagonal Prism Hexagonal Octagonal Decagonal Dodecagonal Prism Prism Prism Prism # of edges at n-gon for # of faces and face # each vertex sides shape name Faces Teachers Copy: Differentiating between Platonic Solids and Prisms The Five Platonic Solids - What makes them so special? These are the only possible regular polyhedra. They are all regular solids. The same number of sides meet at the same angles at each vertex and they have identical polygons that meet at the same angles at each edge. A Platonic solid is also identified as a convex polyhedron because each face lies in a plane that does not intersect the rest of the polyhedron. Hexahedron Tetrahedron Dodecahedron Octahedron Icosahedron # Vertices # Edges # Faces # of edges at each vertex n-gon for sides # of faces and face shape name Tetrahedron 4 6 4 3 3-gon 4 triangular Cube 8 12 6 3 4-gon 6 square Octahedron 6 12 8 4 3-gon 8 triangular Dodecahedron 20 30 12 3 5-gon 12 pentagonal Icosahedron 12 30 20 5 3-gon 20 triangular Prisms: A prism is formed with two parallel congruent polygons (the bases - top and bottom) that are connected at the edges (the bases - top and bottom) with rectangles. Triangular Prism Rectangular Prism Pentagonal Prism # # Vertices Edges Triangular Prism Rectangular Prism Pentagonal Prism Hexagonal Prism Hexagonal Octagonal Decagonal Dodecagonal Prism Prism Prism Prism # of edges at n-gon for # of rectangular # each vertex base faces Faces 6 9 5 3 3-gon 3 8 12 6 3 4-gon 4 10 15 7 3 5-gon 5 12 18 8 3 6-gon 6 Guided Practice: Convex Polyhedra & Euler’s Formula Key Terms: Convex Polygon/Polyhedron: A polygon/polyhedron is convex if every line segment connecting any two points on or within the polygon/polyhedron lies entirely in the polygon’s/polyhedron’s interior. Any polyhedron that is topology a sphere will satisfy Euler’s formula. Euler’s Formula: No. of Vertices – No. of Edges + No. of Faces = 2. Example: A tetrahedron has four vertices, four faces, and six edges. Substituting the numbers in we have: 4-6+4=2 Therefore a tetrahedron satisfies Euler’s formula. Verifying a Polyhedron Satisfies Euler’s Formula: 1) Figure out how many faces (F), edges (E), and vertices (V) the polyhedron has. 2) Write down Euler’s Formula: V–E+F=2 3) Substitute for V, E, and F in the formula. 4) If the left side is equal to 2, then the polyhedron satisfies Euler’s formula. Exercise: Fill in the chart below to verify that the Platonic solids satisfy Euler’s formula. Name: Number of Vertices Number of Faces Number of Edges Tetrahedron Cube Octahedron Dodecahedron Icosahedron Do all the platonic solids satisfy Euler’s formula? Note: Not all polyhedra that satisfy Euler’s formula are convex! V–E+F = Euler’s Formula Satisfied? Guided Practice: Convex Polyhedra & Euler’s Formula (Teacher’s Copy) Key Terms: Convex Polygon/Polyhedron: A polygon/polyhedron is convex if every line segment connecting any two points on or within the polygon/polyhedron lies entirely in the polygon’s/polyhedron’s interior. Any polyhedron that is topology a sphere will satisfy Euler’s formula. Euler’s Formula: No. of Vertices – No. of Edges + No. of Faces = 2. Example: A tetrahedron has four vertices, four faces, and six edges. Substituting the numbers in we have: 4-6+4=2 Therefore a tetrahedron satisfies Euler’s formula. Verifying a Polyhedron Satisfies Euler’s Formula: 1) Figure out how many faces (F), edges (E), and vertices (V) the polyhedron has. 2) Write down Euler’s Formula: V–E+F=2 3) Substitute for V, E, and F in the formula. 4) If the left side is equal to 2, then the polyhedron satisfies Euler’s formula. Exercise: Fill in the chart below to verify that the Platonic solids satisfy Euler’s formula. Name: Number of Vertices Number of Faces Number of Edges V–E+F = Tetrahedron Cube Octahedron Dodecahedron Icosahedron 4 8 6 20 12 4 6 8 12 20 6 12 12 30 30 2 2 2 2 2 Do all the Platonic solids satisfy Euler’s formula? Yes Note: Not every polyhedron that satisfies Euler’s formula is convex. Euler’s Formula Satisfied? Yes Yes Yes Yes Yes Guided Practice: Interior & Exterior Angles of Regular Polygons Key Terms: Interior Angles: To find the total measure of all interior angles in a polygon: 1) Find the number of sides. Let this be n. 2) Plug n into interior angle formula: (n – 2) x 180 3) To find the measure of each interior angle in a regular polygon, divide your answer to number 2 by the number of sides (n). Exterior Angles: To find the total measure of each exterior angle in a regular polygon: 1) Find the number of sides. Let this be n. 2) Plug into the exterior angle formula: Regular Polygon Face of an Icosahedron Octagon Face of a Dodecahedron Hexagon Face of a Cube Nonagon Number of Sides (n) Measure of Each Interior Angle Measure of Each Exterior Angle Guided Practice: Interior & Exterior Angles of Regular Polygons (Teacher’s Copy) Key Terms: Interior Angles: To find the total measure of all interior angles in a polygon: 1) Find the number of sides. Let this be n. 2) Plug n into interior angle formula: (n – 2) x 180 3) To find the measure of each angle in a regular polygon, divide your answer to number 2 by the number of sides (n) Exterior Angles: To find the measure of each exterior angle in a regular polygon: 1) Find the number of sides. Let this be n. 2) Plug into exterior angle formula: Regular Polygon Face of an Icosahedron Number of Sides (n) Measure of Each Interior Angle Measure of Each Exterior Angle 3 60 120 Octagon 8 135 45 Face of a Dodecahedron 5 108 72 Hexagon 6 120 60 Face of a Cube 4 90 90 Nonagon 9 140 40 Finding Surface Area & Volume of Platonic Solids Key Terms: Area of a Polygon: The unique real number assigned to any polygon which indicates the number of non-overlapping square units contained in the polygon’s interior. Surface Area: The sum of the areas of all the faces or curved surfaces of a solid figure. Inradius ( ): For a regular solid, the radius of its inscribed sphere. For each Platonic solid, there is a formula for the inradius in terms of the side length – see table below. Volume: A measure of the number of cubic units needed to fill the space inside a solid figure. -Formula to find the volume of a regular solid: V = f ( )(face area)( ), where f is the number of faces and face area is the area of any face of the regular polyhedron. Fill in the table below: Platonic Solid: Tetrahedron Number of Faces: Length Face of Shape: Base (cm) Length of Inradius (cm) 1 0.20412 1 0.5 Octahedron 1 0.40825 Dodecahedron 1 1.11352 Icosahedron 1 0.75576 Cube Area of One Face: (cm2) Surface Area: (cm2) Volume of Figure: (cm3) Challenge Question: Explain how to find the Area of a face in the Dodecahedron. Finding Surface Area & Volume of Platonic Solids (Teacher’s Copy) Key Terms: Area of a Polygon: The unique real number assigned to any polygon which indicates the number of non-overlapping square units contained in the polygon’s interior. Surface Area: The sum of the areas of all the faces or curved surfaces of a solid figure. Inradius ( ): For a regular solid, the radius of its inscribed sphere. Volume: A measure of the number of cubic units needed to fill the space inside a solid figure. -Formula to find the volume of a regular solid: V = f ( )(face area)( ), where f is the number of faces and face area is the area of any face of the regular polyhedron. Fill in the table below: Length of Inradius: Side: (cm) (cm) Area of One Face: (cm2) Surface Area: (cm2) Volume: (cm3) Platonic Solid: Number of Faces: Face Shape: Tetrahedron 4 Triangle 1 0.20412 √3/4 √3 √2/12 Cube 6 Square 1 0.5 1 6 1 Octahedron 8 Triangle 1 0.40825 √3/4 2√3 √2/3 Dodecahedron 12 Pentagon 1 1.11352 1.720 20.646 7.663 Icosahedron 20 Triangle 1 0.75576 √3/4 5√3 2.182 Challenge Question: Explain how to find the Area of a face in the Dodecahedron. A pentagon consists of 5 triangles. Find the area of one triangle, multiply by 5. Name______________________________ Guided Practice: Platonic Solids and Probability Each pair of students has a collection of polyhedra consisting of the five origami Platonic solids. Fill in the chart below for the number of faces of each color that each Platonic Solid has. Your total in a given column should equal the number of faces for that Platonic solid. Tetrahedron Cube Octahedron Dodecahedron Icosahedron # Red Faces: # Blue Faces: # Yellow Faces: Total: Based on the above chart, calculate the probability of rolling each color. When you add the probabilities in any given column, they should sum to one. Tetrahedron Cube Octahedron Dodecahedron Icosahedron Probability of Rolling Red Probability of Rolling Blue Probability of Rolling Yellow Total: 1 – 4 Circle the event that is more likely to happen. Hint: Use the information that you calculated above. If the likelihood of the two events is the same, write “Equal.” 1. Tetrahedron Landing on Yellow OR Octahedron Landing on Blue 2. Cube Landing on Blue OR Dodecahedron Landing on Red 3. Tetrahedron Landing on Red OR Icosahedron Landing on Yellow 4. Icosahedron Landing on Blue OR Dodecahedron Landing on Red 5 – 8 If we roll two of the Platonic solids, what is the probability that exactly one of them will land on RED? 5. If we roll a Tetrahedron and an Octahedron: ______ □ ______ = ______ □ ______ = ______ If we roll a Dodecahedron and a Cube: ______ □ ______ = ______ If we roll a Tetrahedron and an Icosahedron: ______ □ ______ = ______ 6. If we roll a Cube and an Icosahedron: ______ 7. 8. 9 – 12 If we roll two of the Platonic solids, what is the probability that both of them will land on BLUE? 9. If we roll a Tetrahedron and an Octahedron: ______ □ ______ = ______ □ ______ = ______ 11. If we roll a Dodecahedron and a Cube: ______ □ ______ = ______ 12. If we roll a Tetrahedron and an Icosahedron: ______ □ ______ = ______ 10. If we roll a Cube and an Icosahedron: ______ Roll each Platonic solid and keep track of the number of times you land on each color (Tetrahedron: Roll 16 times, Cube: Roll 24 times, Octahedron: Roll 32 times, Dodecahedron: Roll 50 times, Icosahedron: Roll 80 times) Enter the number of times you land on each color over the total number of rolls for that solid. Your total for any given solid should be ONE: Tetrahedron Red Faces: Blue Faces: Yellow Faces: Total: Cube Octahedron Dodecahedron Icosahedron Review your results. Use some specific comparisons of sample frequencies vs. theoretical probabilities. Be sure to list some reasons why the sample frequencies and theoretical probabilities might differ. Guided Practice: Platonic Solids and Probability- Teacher’s Copy Each pair of students has a collection of polyhedra consisting of the five origami Platonic solids. Fill in the chart below for the quantity of each color that each Platonic solid has. Your total should equal the number of faces each Platonic solid has. Tetrahedron Cube Octahedron Dodecahedron Icosahedron # Red Faces: 2 1 3 4 7 # Blue Faces: 1 2 4 4 5 # Yellow Faces: 1 3 1 4 8 Total: 4 6 8 12 20 Based on the above chart, calculate the probability of rolling each color. When you add the probabilities for the colors they will add to one. Tetrahedron Cube Octahedron Dodecahedron Icosahedron Probability of Rolling Red 1/2 1/6 3/8 1/3 7/20 Probability of Rolling Blue 1/4 1/3 1/2 1/3 ¼ Probability of Rolling Yellow 1/4 1/2 1/8 1/3 2/5 1 1 1 1 1 Total: 1 – 4 Circle the event that is more likely to happen. If the likelihood of the two events is the same, write “Equal”. 1. Tetrahedron Landing on Yellow OR Octahedron Landing on Blue 2. Cube Landing on Blue OR Dodecahedron Landing on Red - Equal 3. Tetrahedron Landing on Red OR Icosahedron Landing on Yellow 4. Icosahedron Landing on Blue OR Dodecahedron Landing on Red 5 – 8 If we roll two of the Platonic solids, what is the probability that exactly one of them will land on RED? 5. 6. 7. 8. If we roll a Tetrahedron and an Octahedron: (1/2)(5/8) + (1/2)(3/8) = 1/2 If we roll a Cube and an Icosahedron: (1/6)(13/20) + (5/6)(7/20) = 48/120 = 2/5 If we roll a Dodecahedron and a Cube: (1/3)(5/6) + (2/3)(1/6) = 7/18 If we roll a Tetrahedron and an Icosahedron: (1/2)(13/20) + (1/2)(7/20) = 1/2 9 – 12 If we roll two of the Platonic solids, what is the probability that both of them will land on BLUE? 9. 10. 11. 12. If we roll a Tetrahedron and an Octahedron: 1/4 × 1/2 = 1/8 If we roll a Cube and an Icosahedron: 1/3 × 1/4 = 1/12 If we roll a Dodecahedron and a Cube: 1/3 × 1/3 = 1/9 If we roll a Tetrahedron and an Icosahedron: 1/4 × 1/2 = 1/8 Roll each Platonic solid and keep track of the number of times you land on each color (Tetrahedron: Roll 16 times, Cube: Roll 24 times, Octahedron: Roll 32 times, Dodecahedron: Roll 50 times, Icosahedron: Roll 80 times) Enter the number of times you land on each color over the total number of rolls for that solid. Your total for any given solid should be ONE: Tetrahedron Cube Octahedron Dodecahedron Icosahedron Red Faces: Blue Faces: Yellow Faces: Total: Review your results. Use some specific comparisons of sample frequencies vs. theoretical probabilities. Be sure to list some reasons why the sample frequencies and theoretical probabilities might differ. DO THE POLYHEDRON Everybody’s doing a brand new shape now Come on baby Do the Polyhedron I know you get to roll it to find your chance now Come on baby do the polyhedron With origami you can do it with ease It’s easier than learning your Kappa Delta Phi’s So come on come on and do this origami with me You’ve gotta crease your edge now Come on baby fold up fold back Well I think you’ve got the knack Now that you can do it let’s make a solid now Come on baby do the polyhedron A fold it, mold it motion like a Dr. Straight now Come on baby do the polyhedron Do it nice and straight now don’t lose control A little bit of algebra and a lot of folds Come on come on do the polyhedron with me Yeah yeah yeah yeah Move around your paper to make your creation (Come on baby, do the polyhedron) Do it Makin’ Solids like you got the notion, (Come on baby, do the polyhedron) There's never been an angle that’s so easy to see, It even makes math history seem like a breeze) So come on, come on and, do the polyhedron with me