Simplices are higher dimensional analogs of line segments and triangle, such as a tetrahedron. We begin this lecture by discussing convex combinations and convex hulls, and showing a natural hierarchy from point to line segment to triangle to tetrahedron. Each of these also has a standard representation as the convex hull of the unit basis vectors in a vector space.

Simplicial complexes are arrangements of simplices where any two are either disjoint or meet at a face, where by face we mean the convex hull of any subset of the vertices of a simplex.

Each simplex has two possible orientations, corresponding to a particular ordering of the vertices. We show how an orientation of a tetrahedron induces orientations on each of its four triangular faces. The boundary of a simplex is then an alternating combination of its oriented faces of dimension one less. A key theorem is that if we apply the boundary to a boundary, we always get zero. This is the basis of the definition of homology.

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Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...

My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.

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Here are the Insights into Mathematics Playlists:

   • MathHistory: A course in the History ...  
   • WildTrig: Intro to Rational Trigonometry  
   • Math Foundations  
   • Wild Linear Algebra  
   • Famous Math Problems  
   • Box Arithmetic  
   • Elementary Mathematics (K-6) Explained  
   • Year9Maths  
   • Ancient Mathematics  
   • Wild West Banking  
   • Sociology and Pure Mathematics  
   • Sociology and Pure Physics  
   • Old Babylonian Mathematics  
   • Probability and Statistics: an introd...  
   • Boole's Logic and Circuit Analysis  
   • Universal Hyperbolic Geometry  
   • Differential Geometry  
   • Algebraic Topology  
   • MathSeminars  
   • Playing Go  
   • Diffusion Symmetry: A bridge between ...  

Here are the Wild Egg Maths Playlists (some available only to Members!)

   • Algebraic Calculus One  
   • Classical to Quantum (Members only)  
   • Algebraic Calculus Two  
   • Advice for mathematics exploration (M...  
   • Solving Polynomial Equations (Members...  
   • The Hexagrammum Mysticum (Members only)  
   • Algebraic Calculus and dCB curves (Me...  
   • Dynamics on Graphs (Members only)  
   • Playlist  
   • Triangle Geometry  
   • Explorations with q-series (Members o...  
   • Six: An elementary course in Pure Mat...  
   • Maxel Inverses and Orthogonal Polynom...  
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