Stereographic projection is also applied to the visualization of polytopes. In a Schlegel diagram, an -dimensional polytope in is projected onto an -dimensional sphere, which is then stereographically projected onto . The reduction from to can make the polytope easier to visualize and understand.
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The fibers of the Hopf fibration stereographically project to a family of Villarceau circles in R3. The Hopf fibration has many implications, some purely attractive, others deeper. For example, stereographic projection S3 → R3 induces a remarkable structure in R3, which in turn illuminates the topology of the bundle . Stereographic projection preserves circles and maps the Hopf fibers to geometrically perfect circles in R3 which fill space.
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A 4D clifford torus, stereographically projected into 3D, looks like a torus. A double rotation can be seen as a helical path. In 4D, a double rotation symmetry can be generated as the composite of two orthogonal rotations.Charles Howard Hinton (1906) The Fourth Dimension (Google eBook) S. Sonnenschein & Company p.223 It is similar to 3D screw axis which is the composite of a rotation and an orthogonal translation.
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Then v(p) is the displacement vector of this projected point relative to p. According to the hairy ball theorem, there is a p such that v(p) = 0, so that s(p) = p. This argument breaks down only if there exists a point p for which s(p) is the antipodal point of p, since such a point is the only one that cannot be stereographically projected onto the tangent plane of p.
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De plana spera, geometric drawing This treatise of five propositions deals with various aspects of stereographic projection (used in planispheric astrolabes). The first and historically the most important proposition proves for all cases that circles on the surface of a sphere when projected stereographically on a plane remain circles (or a circle of infinite radius, i.e., a straight line). While this property was known long before Jordanus, it had never been proved.
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In 2014, Hart founded a research group called eleVR, with Emily Eifler and Andrea Hawksley, to research virtual reality (VR). The group created VR videos, and had also collaborated on educational computer games. They created the game Hypernom, where the player has to eat part of 4 dimensional polytopes which are stereographically projected into 3D and viewed using a virtual reality headset. In June, eleVR released an open source web video player that worked with the Oculus Rift.
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The dial of the Zytglogge's astronomical clock is built in the form of an astrolabe. It is backed by a stereographically projected planisphere divided into three zones: the black night sky, the deep blue zone of dawn and the light blue day sky. The skies are crisscrossed with the golden lines of the horizon, dawn, the tropics and the temporal hours, which divide the time of daylight into twelve hours whose length varies with the time of year.Bellwald (1983), 19.
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However, one can approximately visualize it as a disk, as follows. Any line through the origin intersects the southern hemisphere ≤ 0 in a point, which can then be stereographically projected to a point on a disk. Horizontal lines intersect the southern hemisphere in two antipodal points along the equator, either of which can be projected to the disk; it is understood that antipodal points on the boundary of the disk represent a single line. (See quotient topology.) So any set of lines through the origin can be pictured, almost perfectly, as a set of points in a disk.
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Yet this figure of 2n circles shows beyond a doubt the superiority of Cox's chain over Clifford's; for the latter is included as a special case when half the circles in the former shrink into points. Cox's plane figure of 2n circles can be derived by elementary methods.Herbert W. Richmond (1941) "On a chain of theorems due to Homersham Cox", Journal of the London Mathematical Society 16: 105–7, H. S. M. Coxeter derived Clifford's theorem by exchanging the arbitrary point on a line ab with an arbitrary sphere about 0 which then intersects ab. The planes a, b, c, ... intersect this sphere in circles which can be projected stereographically into a plane.
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Tesseract, in stereographic projection, in double rotation A 4D Clifford torus stereographically projected into 3D looks like a torus, and a double rotation can be seen as in helical path on that torus. For a rotation whose two rotation angles form a rational number, the paths will eventually reconnect, while for an irrational ratio they will not. An isoclinic rotation will form a Villarceau circle on the torus, while a simple rotation will form a circle parallel or perpendicular to the central axis. For each rotation of 4-space (fixing the origin), there is at least one pair of orthogonal 2-planes and each of which is invariant and whose direct sum is all of 4-space.
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