''The Geometry of the Octonions'' is a mathematics book on the
octonions, a system of numbers generalizing the
complex numbers and
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s, presenting its material at a level suitable for undergraduate mathematics students. It was written by
Tevian Dray
Tevian Dray (born March 17, 1956) is an American mathematician
who has worked in general relativity, mathematical physics,
geometry, and both science and mathematics education. He was elected a Fellow of the American Physical Society in 2010.
H ...
and
Corinne Manogue
Corinne Alison Manogue (born March 3, 1955) is an American physicist who has worked in general relativity, mathematical physics, and physics education. She was elected a Fellow of the American Physical Society in 2005, and was an inaugural Fellow ...
, and published in 2015 by World Scientific. The Basic Library List Committee of the
Mathematical Association of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...
has suggested its inclusion in undergraduate mathematics libraries.
Topics
The book is subdivided into three parts, with the second part being the most significant. Its contents combine both a survey of past work in this area, and much of its authors' own researches.
The first part explains the
Cayley–Dickson construction, which constructs the
complex numbers from the
real numbers, the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s from the complex numbers, and the octonions from the quaternions. Related algebras are also discussed, including the
sedenions (a 16-dimensional real algebra formed in the same way by taking one more step past the octonions) and the split real
unital composition algebras (also called Hurwitz algebras). A particular focus here is on interpreting the multiplication operation of these algebras in a geometric way. Reviewer Danail Brezov notes with disappointment that
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
s, although very relevant to this material, are not covered.
The second part of the book uses the octonions and the other
division algebras associated with it to provide concrete descriptions of the
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s of geometric symmetries. These include
rotation group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
s,
spin groups,
symplectic groups, and the
exceptional Lie groups
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
, which the book interprets as octonionic variants of classical Lie groups.
The third part applies the octonions in geometric constructions including the
Hopf fibration and its generalizations, the
Cayley plane, and the
E8 lattice. It also connects them to problems in physics involving the four-dimensional
Dirac equation, the
quantum mechanics of relativistic
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s,
spinors, and the formulation of quantum mechanics using
Jordan algebras. It also includes material on octonionic
number theory, and concludes with a chapter on the
Freudenthal magic square and related constructions.
Audience and reception
Although presented at an undergraduate level, ''The Geometry of the Octonions'' is not a textbook: its material is likely too specialized for an undergraduate course, and it lacks exercises or similar material that would be needed to use it as a textbook. Readers should be familiar with
linear algebra, and some experience with
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s would also be helpful. The later chapters on applications in physics are heavier going, and require familiarity with quantum mechanics.
The book avoids a proof-heavy formal style of mathematical writing, so much so that reviewer Danail Brezov writes that at points it "seems to lack mathematical rigor".
Related reading
Multiple reviewers suggest that this work would make a good introduction to the octonions, as a stepping stone to the more advanced material presented in other works on the same topic. Their suggestions include the following:
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References
{{DEFAULTSORT:Geometry of the Octonions, The
Mathematics books
2015 non-fiction books
Octonions