Visualization of a 3D Levi-Civita symbol
Posted on Sun 08 December 2019 in Blog
The Levi-Civita symbol is that puzzling \epsilon_{ijk} term that appears in the calculation of determinants and vector products. It has, thus, lots of applications in algebra, geometry, mechanics, electromagnetism, ... A (boring) definition is given in the textbox below.
The value of \epsilon_{ijk} goes as follows:
It is 0 if the value of any index is repeated, such as in ijk = 112
It is 1 if the indices are ijk \in123, 312, 231
It is -1 if the indices are ijk \in 321, 132, 213
I am interested in visualization. Unfortunately, the Levi-Civita symbol has three indices, i, j and k, so we cannot represent it as a matrix or a vector. But, taking advantage of its discrete nature (it can only take 3 different values) we can create a "cubic matrix" to visualize it. This matrix contains 3 \times 3 \times 3 cubes, each of them univocally identified by a triplet ijk, and the color tells us its value (grey for 0, green for 1, red for -1).
That's what I did in a GeoGebra applet. I leave it here, just in case anyone finds it interesting. Click on the image to open an interactive version where the cube can be rotated.