# Metrics

The *metric* used in the 5D conformal geometric algebra over 3D Euclidean space is the $$5 \times 5$$ matrix $$\mathfrak g$$ given by

- $$\mathfrak g = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & -1 & 0\\\end{bmatrix}$$ .

The *metric exomorphism matrix* $$\mathbf G$$, often just called the "metric" itself, corresponding to the metric $$\mathfrak g$$ is the $$32 \times 32$$ matrix shown below.

The *metric antiexomorphism matrix* $$\mathbb G$$, often called the "antimetric", corresponding to the metric $$\mathfrak g$$ is the negation of the metric exomorphism matrix $$\mathbf G$$.

The metric and antimetric determine duals, dot products, and geometric products.