Editing 2735: Coordinate Plane Closure
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Closure can also be used in another sense, relating to the topology of a set; roughly speaking, a description of what parts of the set are "close" to others. In this sense, if one takes the closure of a plane with a hole, the result is indeed an intact plane, provided the hole is sufficiently (infinitesimally) small. | Closure can also be used in another sense, relating to the topology of a set; roughly speaking, a description of what parts of the set are "close" to others. In this sense, if one takes the closure of a plane with a hole, the result is indeed an intact plane, provided the hole is sufficiently (infinitesimally) small. | ||
− | The title text notes that 3D graphs that cross the relevant x and y coordinates, but with non-zero z coordinates whilst in that zone, should be fine, since the hole only exists in the plane where z = 0. However, if they pass close - i.e. the z coordinate is small in this region - they should be wary of two dimensional graph lines suddenly becoming three-dimensional and interfering with them. This could be because they have intentionally entered three-dimensional space to avoid the closure, or possibly they have inadvertently been 'launched' above/below the plane by the torn and warped edges of the surface | + | The title text notes that 3D graphs that cross the relevant x and y coordinates, but with non-zero z coordinates whilst in that zone, should be fine, since the hole only exists in the plane where z = 0. However, if they pass close - i.e. the z coordinate is small in this region - they should be wary of two dimensional graph lines suddenly becoming three-dimensional and interfering with them. This could be because they have intentionally entered three-dimensional space to avoid the closure, or possibly they have inadvertently been 'launched' above/below the plane by the torn and warped edges of the surface. This is similar to warnings to road traffic in open lanes being warned of traffic merging from lanes that have been closed due to works or any other general warning of increased congestion upon a parallel route used as a diversion. |
The concept of 2D objects suddenly entering 3D space — in a way that creates interesting drama and conflict — is the subject of a book, Flatland, that Randall has referenced repeatedly, such as in [[721: Flatland]]. | The concept of 2D objects suddenly entering 3D space — in a way that creates interesting drama and conflict — is the subject of a book, Flatland, that Randall has referenced repeatedly, such as in [[721: Flatland]]. |