The octonions are the largest of the four normed division algebras: the real numbers ($\mathbb{R}$), complex numbers ($\mathbb{C}$), quaternions ($\mathbb{H}$), and octonions ($\mathbb{O}$). While the octonions are nonassociative, they play a significant role in various areas of mathematics, including Clifford algebras, spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and exceptional Lie groups. The octonions were discovered by John T. Graves, a friend of Hamilton, who found them while exploring the properties of quaternions. Hamilton's discovery of quaternions was motivated by his quest to find a 3-dimensional normed division algebra, which he eventually achieved. However, the octonions, being 8-dimensional, are nonassociative and thus lack the geometric and physical applications that quaternions have. The octonions are constructed using the Cayley-Dickson construction, which builds new algebras from existing ones. This construction explains why quaternions are noncommutative and octonions are nonassociative. The octonions can also be described using the Fano plane, a geometric structure that helps visualize their multiplication rules. Additionally, the octonions are related to Clifford algebras, which are associative algebras generated by anticommuting square roots of -1. The Clifford algebra $\mathrm{Cliff}(n)$ is a representation of the normed division algebra $\mathbb{O}$, and this relationship is crucial for understanding the geometrical significance of the octonions. The octonions are also connected to spinors and triality, which are important concepts in theoretical physics and mathematics. Spinors are representations of the Clifford algebra $\mathrm{Cliff}(n-1)$, and the relation between spinors and vectors in $n$ dimensions gives rise to normed triality. This triality is a trilinear map that defines a division algebra in dimensions 1, 2, 4, and 8. The octonions are the only normed division algebra in these dimensions, and they play a central role in the theory of exceptional Lie groups and string theory. In conclusion, the octonions, though less well-known than quaternions, are fundamental in advanced mathematics and physics, particularly in the study of rotations, spinors, and the structure of spacetime. Their nonassociativity and dimensionality make them unique and intriguing, and their connections to other mathematical structures continue to be explored and studied.The octonions are the largest of the four normed division algebras: the real numbers ($\mathbb{R}$), complex numbers ($\mathbb{C}$), quaternions ($\mathbb{H}$), and octonions ($\mathbb{O}$). While the octonions are nonassociative, they play a significant role in various areas of mathematics, including Clifford algebras, spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and exceptional Lie groups. The octonions were discovered by John T. Graves, a friend of Hamilton, who found them while exploring the properties of quaternions. Hamilton's discovery of quaternions was motivated by his quest to find a 3-dimensional normed division algebra, which he eventually achieved. However, the octonions, being 8-dimensional, are nonassociative and thus lack the geometric and physical applications that quaternions have. The octonions are constructed using the Cayley-Dickson construction, which builds new algebras from existing ones. This construction explains why quaternions are noncommutative and octonions are nonassociative. The octonions can also be described using the Fano plane, a geometric structure that helps visualize their multiplication rules. Additionally, the octonions are related to Clifford algebras, which are associative algebras generated by anticommuting square roots of -1. The Clifford algebra $\mathrm{Cliff}(n)$ is a representation of the normed division algebra $\mathbb{O}$, and this relationship is crucial for understanding the geometrical significance of the octonions. The octonions are also connected to spinors and triality, which are important concepts in theoretical physics and mathematics. Spinors are representations of the Clifford algebra $\mathrm{Cliff}(n-1)$, and the relation between spinors and vectors in $n$ dimensions gives rise to normed triality. This triality is a trilinear map that defines a division algebra in dimensions 1, 2, 4, and 8. The octonions are the only normed division algebra in these dimensions, and they play a central role in the theory of exceptional Lie groups and string theory. In conclusion, the octonions, though less well-known than quaternions, are fundamental in advanced mathematics and physics, particularly in the study of rotations, spinors, and the structure of spacetime. Their nonassociativity and dimensionality make them unique and intriguing, and their connections to other mathematical structures continue to be explored and studied.