We are sometimes given a vagina—and that designates a “woman”—virgin, bride, etc.—and sometimes a penis—and that indicates a “man”—bachelor, groom, etc. This physiological accident was never anything more than the effect of an assuredly ironic causality: the laws of Euclidian geometry. In a four-dimensional study … vagina and penis, like an anamorphic illusion, would immediately lose all distinctive character. It is the same object that we would sometimes see as “male” and sometimes as “female,” in this perfect mirror-like reversal of the body that presupposes, because it takes place, the existence of a fourth dimension.
—Jean Clair, Sur Marcel Duchamp et la Fin de l’Art
An object that has a length has one dimension. A length and width make two, and an object with a length, a width, and a height has three dimensions. Any object that really exists, from the time it came into existence until the moment it vanishes forever, has duration, the fourth dimension. Before the fourth dimension began to be treated as time, however, it was briefly described as a transcendental dimension of space and imagined as the domain of whatever way of being in extension came after height.
The possibility of higher-dimensional space is often traced to the twenty-four-year old Immanuel Kant, who speculated, in Thoughts on the True Estimation of Living Forces (1747), that “if it is possible that there are extensions with other dimensions, it is also very probable that God has somewhere brought them into being; for His works have all the magnitude and manifoldness of which they are capable.” He later returned to this thought in The Prolegomena to Any Future Metaphysics (1783), where, in a puzzling little paragraph, he asks himself: If all space were empty but for a single human hand, would it make sense to ask whether that hand was specifically a right hand?
To visualize the problem Kant posed, imagine the outline of a hand printed onto a transparent surface. It can appear to be either a left or a right hand, depending on the position of the observer. Left or right only make sense within the boundaries of two-dimensional space; once you move into the third dimension, left and right become observer-dependent, rather than independent, characteristics. Their outline is neither left nor right, but sometimes left and sometimes right.
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