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→Quaternionic projective plane: later rediscovered by Edward Witten and Michael Atiyah. |
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:[''cq''<sub>0</sub>:''cq''<sub>1</sub>: ... :''cq''<sub>''n''</sub>].
In the language of [[group action]]s, '''H'''''P''<sup>''n''</sup> is the [[orbit space]] of '''H'''<sup>''n''+1</sup> by the action of '''H'''*, the multiplicative group of non-zero quaternions. By first projecting onto the unit sphere inside '''H'''<sup>''n''+1</sup> one may also regard '''H'''''P''<sup>''n''</sup> as the orbit space of ''S''<sup>4''n''+3</sup> by the action of Sp(1), the group of unit quaternions. The sphere ''S''<sup>4''n''+3</sup>
:<math>\mathrm{Sp}(1) \to S^{4n+3} \to \mathbb HP^n.</math>
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