It not only generalizes the classic vector algebra to enable construction of (unlocated) bivectors, trivectors and multivectors, it is also an algebra par excellence for working with located entities such as points, lines, planes and multiplanes.

But multiplanes are not alone in their space. To every multiplane corresponds a docked multi-sphere and vice versa. (A docked multisphere passes through the origin.) Corresponding points on a multiplane-multisphere pair are inverses. And because we can easily dock a multisphere by adding a displacement vector to its points, we can work with multispheres by operating on their corresponding multiplanes. For example: we can intersect two multispheres, or a multisphere and a multiplane; construct the best-fit multisphere to a system of points; compute the complex of circles for a Clifford circle theorem, or generate the in-multisphere of a simplex.

The book comprises twenty chapters, structured into five groups:

• six chapters exploring the geometry of multiplanes,

• five chapters exploring the geometry of multispheres,

• seven chapters involving shorter case studies on multiplanes and multispheres,

• a longer case study applying lines and circles to Clifford’s circle theorems in the plane,

• a longer case study applying multiplanes and multispheres to simplexes inn-space.

Nearly every chapter features specialized Grassmann Algebra-Mathematica functions for doing the requisite computations. Because an understanding of how these functions work is best given by example, I have integrated comments on their usage into the text. A function name will always be given in Mathematica’s default coding font so a scan of the index or table of contents will find its first use.

This Book has some diagrams printed in colour and is more expensive to print. It normally retails at a higher price than its companion volumes.

Contents Overview

1 An introduction to multiplanes 1

2 Multiplanes 21

3 The Triangle Formula 28

4 Orthogonal points of multiplanes 37

5 Orthonormalization of multiplanes 48

6 Operations on points and multiplanes 56

7 An introduction to multispheres 64

8 Converting multisphere specifications 72

9 The inverse of a multisphere 87

10 Intersections of multiplanes and multispheres 97

11 Intersections of multispheres 109

12 Fitting multiplanes 125

13 Fitting multispheres 154

14 Closest points 169

15 Hyperplanes 184

16 Docked multispheres 195

17 Intersections of docked multispheres 216

18 Tangent multiplanes 225

19 Clifford’s circle theorems 231

20 Simplexes 278

Coda 352