Western Oregon University Digital Commons@WOU Academic Excellence Showcase Proceedings Student Scholarship 2018-06-01 Classifying Regular Polytopes in Dimension 4 and Beyond Brittany Johnson Western Oregon University, [email protected] Follow this and additional works at: https://digitalcommons.wou.edu/aes Part of the Physical Sciences and Mathematics Commons Recommended Citation Johnson, Brittany, "Classifying Regular Polytopes in Dimension 4 and Beyond" (2018). Academic Excellence Showcase Proceedings. 115. https://digitalcommons.wou.edu/aes/115 This Presentation is brought to you for free and open access by the Student Scholarship at Digital Commons@WOU. It has been accepted for inclusion in Academic Excellence Showcase Proceedings by an authorized administrator of Digital Commons@WOU. For more information, please contact [email protected], [email protected], [email protected]. Introduction Polygons, Polyhedra, Polychora Higher Dimension Polytopes Classifying Regular Polytopes in Dimension 4 and Beyond Brittany Johnson Western Oregon University May 31, 2018 Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond How many regular convex polytopes are there in each dimension? How can we prove this? Introduction Polygons, Polyhedra, Polychora Motivating Questions Higher Dimension Polytopes Motivating Questions Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond How can we prove this? Introduction Polygons, Polyhedra, Polychora Motivating Questions Higher Dimension Polytopes Motivating Questions How many regular convex polytopes are there in each dimension? Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Introduction Polygons, Polyhedra, Polychora Motivating Questions Higher Dimension Polytopes Motivating Questions How many regular convex polytopes are there in each dimension? How can we prove this? Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polygons Definition A polygon is a closed and connected shaped bounded by a finite number of lines. Polygons Not Polygons Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polygons Definition A regular convex polygon is a polygon that is equilateral, equiangular, and whose interior forms a convex set. Regular Convex Polygons Not Regular Convex Polygons Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond All sides are the same lengths The interior angles are all congruent They can be represented by a Schl¨aflisymbol This is of the form {p} where p represents the number of sides It is unique {3} {4} {5} {6} {7} (we can keep going!) There are infinitely many! Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polygons Important things to notice: Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond The interior angles are all congruent They can be represented by a Schl¨aflisymbol This is of the form {p} where p represents the number of sides It is unique {3} {4} {5} {6} {7} (we can keep going!) There are infinitely many! Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polygons Important things to notice: All sides are the same lengths Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond They can be represented by a Schl¨aflisymbol This is of the form {p} where p represents the number of sides It is unique {3} {4} {5} {6} {7} (we can keep going!) There are infinitely many! Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polygons Important things to notice: All sides are the same lengths The interior angles are all congruent Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond There are infinitely many! Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polygons Important things to notice: All sides are the same lengths The interior angles are all congruent They can be represented by a Schl¨aflisymbol This is of the form {p} where p represents the number of sides It is unique {3} {4} {5} {6} {7} (we can keep going!) Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polygons Important things to notice: All sides are the same lengths The interior angles are all congruent They can be represented by a Schl¨aflisymbol This is of the form {p} where p represents the number of sides It is unique {3} {4} {5} {6} {7} (we can keep going!) There are infinitely many! Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond ANSWER: The regular convex polyhedra, or the Platonic solids Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra QUESTION: What is the 3-dimensional analog of the 2-dimensional regular convex polygons? Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra QUESTION: What is the 3-dimensional analog of the 2-dimensional regular convex polygons? ANSWER: The regular convex polyhedra, or the Platonic solids Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Notice that the faces are a regular polygon! All sides are the same All faces are congruent lengths Interior angles are congruent Angles formed by faces are congruent Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra (The Platonic Solids) Square Cube Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Notice that the faces are a Interior angles are regular polygon! congruent Angles formed by faces are congruent Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra (The Platonic Solids) Square Cube All sides are the same All faces are congruent lengths Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Interior angles are congruent Angles formed by faces are congruent Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra (The Platonic Solids) Square Cube All sides are the same All faces are congruent lengths Notice that the faces are a regular polygon! Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra (The Platonic Solids) Square Cube All sides are the same All faces are congruent lengths Notice that the faces are a Interior angles are regular polygon! congruent Angles formed by faces are congruent Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond { 4 , 3 } {4} {3} Square facet Triangle vertex figure Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra (The Platonic Solids) { p , q } Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond {4} {3} Square facet Triangle vertex figure Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra (The Platonic Solids) { p , q } { 4 , 3 } Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond {3} Triangle vertex figure Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra (The Platonic Solids) { p , q } { 4 , 3 } {4} Square facet Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra (The Platonic Solids) { p , q } { 4 , 3 } {4} {3} Square facet Triangle vertex figure Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond Introduction Polygons Polygons, Polyhedra, Polychora Polyhedra Higher Dimension Polytopes Polychora Regular Convex Polyhedra (The Platonic Solids) Definition An n-polytope’s facets are the type of (n − 1)-polytopes that bound it. Definition The vertex figure of an n-polytope is the (n − 1)-dimensional convex hull formed by connecting the center of each of the (n − 2)-elements that are incident on a given vertex. Brittany Johnson Western Oregon University Classifying Regular Polytopes in Dimension 4 and Beyond In other words, the sum of the interior angles that meet