This video explores symmetrical component theory in a way that is not presented in electrical engineering school. Starting with the history and building up 1 phase at a time, the math of symmetrical components is explained in a visual way, showing the connection back to basic geometry. Then the general behavior of symmetrical components for any number of phases is briefly reviewed. To dive deeper into symmetrical component theory, I recommend visiting the Geogebra simulators linked below and interacting with the simulations there. If Symmetrical Component theory has always seemed like a mystery, then manipulating these simulators can help build a more natural intuition for the topic.

Submitted for the Summer of Math Exposition Round 2.

Content by Nathan Kassees. Credit to Oscar Flores for making a lot of the animations here. Check out his page at https://manim.dev/@reduccionista.

Geogebra Simulators
Wave-Phasor Relationship - https://www.geogebra.org/m/kznambqh
Transform system phasors into symmetrical components - https://www.geogebra.org/m/c3dfnj3d
Transform symmetrical components into system phasors - https://www.geogebra.org/m/yxaq6bck
V0 and the triangle centroid - https://www.geogebra.org/m/zkbnwvdv
V1 and the Outer Napoleon Triangle - https://www.geogebra.org/m/jdza87bg
V2 and the Inner Napoleon Triangle - https://www.geogebra.org/m/vthbgtxy
6-Phase Symmetrical Components - https://www.geogebra.org/m/chntv6kh
12-Phase Symmetrical Components - https://www.geogebra.org/m/amarfxe8
1-root(n)-root(n) Napoleon Triangles - https://www.geogebra.org/m/ncqwvj68

For more background on this geometric method of finding symmetrical components, I recommend the textbook chapter below and the references therein:
“Chapter XIII: Determination of Sequence Quantities from Phase Quantities.” Symmetrical Components, as Applied to the Analysis of Unbalanced Electrical Circuits, by C. F. Wagner and R. D. Evans, Mcgraw-Hill, 1933.