Whenever one studies geometry, close attention must be paid to notions of parallelism. When formulating his Elements, Euclid somehow had the foresight to realize that he must postulate his ideas on parallels, for they cannot be deduced from the other axioms. The generations of mathematicians that followed would struggle to prove Euclidıs parallel postulate, but none would succeed. Noteworthy attempts were made by Ptolemy, Proclus, Nasiraddin at-Tusi, John Wallis, Gerolamo Saccheri, Lambert, and Legendre. Once mathematicians began to consider Euclidıs first four axioms along with an alternat ive fifth axiom--describing different behavior of parallels--non-Euclidean geometries were born. The three most instrumental men in this area were Carl Friedrich Gauss, Janos Bolyai, and Nicolai Ivanovich Lobachevsky. The revolutionary nature of the work of these men cannot b e understated, for they truly changed the way mathematicians think about geometry at the most basic level.
The geometry contained within this paper is intended to be built up from Euclidean principles alone. For this reason, I will set the more modern masters aside to give an introduction into the ideas of parallelism in four dimensional space, based on th e Euclidean parallel postulate. One fascinating concept that arises is the idea hyperplanes can be parallel to one another (see Theorems 4-6 below). Henry Parker Manning wrote a very clear book that extends Euclidean notions to four dimensions, so I will take advantage of his concise wording by presenting his section on parallels. The implications of these notions go far beyond simply the words of the theorems. The reader is encouraged to be imaginative and truly contemplate the meaning behind these theorems, for although they are analogous to three-dimensional theorems, they do say quite a bit more.
AXIOM. Through any point not a point of a given line passes one and only one line that lies in a plane with the given line and does not intersect it.
126. Parallel lines and parallel planes. Lines and planes are parallel to one another as in the ordinary geometry: two lines when they lie in one plane and do not intersect, a line and a plane or two planes when they lie in one hyperpl ane and do not intersect.
THEOREM 1. Two lines perpendicular to the same hyperplane are parallel (see Art. 39, Th. 2).
THEOREM 2. A hyperplane perpendicular to one of two parallel lines is perpendicular to the other.
THEOREM 3. If two planes through a point are parallel to a given line they intersect in a parallel line.
THEOREM 4. If a hyperplane intersects one of two parallel planes and does not contain it, the hyperplane intersects the other plane also, and the two lines of intersection are parallel.
For the hyperplane intersects the hyperplane of the parallel planes in a plane which intersects the parallel planes in parallel lines.
THEOREM 5. If a plane meets one of two parallel planes in a single point, it will meet the other in a single point.
THEOREM 6. Two planes absolutely perpendicular to a third are parallel (see Art. 45, Th. ).
THEOREM 7. A plane absolutely perpendicular to one of two parallel planes is absolutely perpendicular to the other.
THEOREM 8. Two planes parallel to a third are parallel to each other.
for a plane absolutely perpendicular to the third is absolutely perpendicular to the first two, and they are parallel by Th. 6.
THEOREM 9. If three parallel planes all intersect a given line, they all lie in one hyperplane.
THEOREM 10. Two planes absolutely perpendicular to two parallel planes are parallel, and two planes parallel respectively t two absolutely perpendicular planes are absolutely perpendicular.
THEOREM 11. If two planes intersect in a line, planes through any point parallel to them intersect in a parallel line and form dihedral angles equal to the dihedral angles formed by the two given planes.
The parallel planes are parallel to the line of intersection of the two given planes, and therefore intersect in a parallel line, by Th. 3. Now a hyperplane perpendicular to these parallel lines (see Th. 2) cuts the planes in lines which contain the s ides of the plane angles of the various dihedral angles formed about the two parallel lines. Corresponding plane angles, and therefore corresponding dihedral angles 1, are eq ual.
COROLLARY. If two planes are perpendicular, planes through any point parallel to them are also perpendicular.
THEOREM 12. If two planes have a point in common, parallel planes through any other point make the same angles .
PROOF. Let A and B be the two given planes having a point O in common, and let A' and B' be planes through a second point O' parallel respectively to A and B. The planes through O' parall el to the common perpendicular planes of A and B are themselves common perpendicular planes of A' and B'. (Th. 11, Cor.). On each of these common perpendicular planes the same angles are cut out as on the corresponding planes at O, since the intersection of any two planes intersecting in a line at O' is parallel to the intersection of the parallel planes at O, and two intersecting line at O' lie in a hyperplane with the parallel lines at O, f orming angles equal to the angles formed by the latter.
COROLLARY. A plane isocline to one of two parallel planes is isocline to the other and makes the same angle with both.
THEOREM 13. Two lines not in the same plane have only one common perpendicular line .
Since the two lines lie in a hyperplane this is always a theorem of geometry of three dimensions, and is proved as in the text-books.
THEOREM 14. If a line and plane do not lie in one hyperplane, they have only one common perpendicular line.
127. Half-parallel planes. Two planes which do not lie in one hyperplane and do not intersect are said to be half-parallel or semi-parallel.
THEOREM 1. The linear elements of two half-parallel planes are all parallel to one another.
THEOREM 2. The linear elements which lie in one of two half-parallel planes are parallel to the other plane, and these are the only lines which lie in one plane and are parallel to the other.
THEOREM 3. Through any point passes one and only one hyperplane perpendicular to each of two half-parallel planes.
THEOREM 4. Two half-parallel planes have one and only one common perpendicular plane.
PROOF. There is one such plane, by Th. 3 of Art. 63. Suppose, then, we have given a plane perpendicular to each of two half-parallel planes. It will intersect these planes in linear elements, the edges of various right dihedral angles, each with one face in the perpendicular plane and one in one of the half-parallel planes. A perpendicular hyperplane intersects the planes in lines which contain the sides of the plane angl es of these dihedral angles, that is, it intersects the perpendicular plane in the common perpendicular line of the lines in which it intersects the two half-parallel planes. There is only one such common perpendicular line, and the given plane is the pl ane determined as in Art. 63 by this common perpendicular line and the linear elements which it intersects.
THEOREM 5. The only common perpendicular lines of two half-parallel planes are those which lie in the common perpendicular plane.
The perpendicular distance or simply the distance, between two half-parallel planes is the distance between the points where they are cut by a common perpendicular line. It is the same for all of these lines, since the common perpendicul ar plane cuts the given plane in parallel lines (Th. 1).
THEOREM 6. The perpendicular distance between two half-parallel planes is less than the distance measured along any line which intersects both and is not perpendicular to both.
PROOF. The perpendicular distance between two elements lying one in each of the two given planes is the distance measured along some line between the intersections of the given planes and this hyperplane. It is less than the distance between the two elements along any line which does not lie in a perpendicular hyperplane. But the intersections of the given plane and the perpendicular hyperplane have for common perpendicular only the line in this hyperplane which is perpendicular to the two given pla nes. Therefore the perpendicular distance between the two given planes is the perpendicular distance between these two intersections, and is less than the distance between the two planes measured on any line that is not perpendicular to both.
THEOREM 7. Two planes through a point parallel respectively to two half-parallel planes intersect in a line which is parallel to their linear elements.
For the line through the point parallel to the linear elements is parallel to the two given planes, and therefore lies in both of the two planes which are parallel to them through the point.
THEOREM 8. If a plane distinct from each of two parallel planes intersects one in a line and does not intersect the other in a line, it will be half-parallel to the second.
PROOF. If the given plane were in a hyperplane with the second parallel plane, this hyperplane, containing the line in which the given plane intersects the first parallel plane, must be the hyperplane of the parallel planes; or if the given plane inte rsected the second parallel plane in a point, it would lie entirely in the hyperplane of the parallel planes. Thus, in either case, we should have a plane lying in the hyperplane of the two parallel planes, intersecting one in a line, and therefore the o ther in a line. As the given plane does not intersect the second parallel plane in a line, it cannot lie in a hyperplane with it nor intersect it at all. They must, therefore, be half-parallel.
THEOREM 9. If a plane perpendicular to one of two absolutely perpendicular planes does not contain their point of intersection, it is half-parallel to the other.
PROOF. Let A and A' be two absolutely perpendicular planes intersecting in a point O, and let B be a plane perpendicular to A but not containing O. Then B cannot lie in a hyperplane with A', for such a hyperplane would intersect A only in a line through O. Nor can B intersect A' even in a point, for then it would contain the line through such a point perpendicular to A, and so contain the point O. B is therefore half-parallel to A'.
128. Lines and planes parallel to a hyperplane. Parallel hyperplanes. A line and a hyperplane, a plane and a hyperplane, or two hyperplanes, are parallel when they do not intersect.
THEOREM 1. If a line, not a line of a given hyperplane, is parallel to a line of the hyperplane, it is parallel to the hyperplane; and if a plane, not a plane of a given hyperplane, is parallel to a plane of the hyperplane, it is parallel to the hype rplane.
THEOREM 2. If a line is parallel to a hyperplane, it is parallel to the intersection of the hyperplane with any plane through it or with any hyperplane through it; and if a plane is parallel to a hyperplane, it is parallel to the intersection of the h yperplane with any hyperplane through it.
THEOREM 3. If a line is parallel to a hyperplane, a line through any point of the hyperplane parallel to the given line lies wholly in the hyperplane; and if a plane is parallel to a hyperplane, a plane or line through any point of the hyperplane para llel to the given plane lies wholly in the hyperplane.
THEOREM 4. Two hyperplanes perpendicular to the same line are parallel.
THEOREM 5. If one of two parallel hyperplanes is perpendicular to a line, the other is also perpendicular to the line.
THEOREM 6. Through a point, not a point of a given hyperplane, can be passed one and only one parallel hyperplane.
In general, we can pass through a point a hyperplane parallel to a given hyperplane, to a given line and plane, or to three given lines; through a line, a hyperplane parallel to a given plane or to two given lines; through a plane, a hyperplane paralle l to a given line. In some cases, however, the construction will give us a hyperplane containing some or all of the given figures, and in some cases more than one hyperplane can be obtained.
THEOREM 7. All the lines and planes in one of two parallel hyperplanes are parallel to the other, and all the lines and planes through a point, parallel to a hyperplane, lie in a parallel hyperplane.
THEOREM 8. If a plane intersects two parallel hyperplanes, or if a hyperplane intersects two parallel planes, the lines of intersection are parallel; and if a hyperplane intersects two parallel hyperplanes, the planes of intersection are parallel.
THEOREM 9. If three non-coplanar lines through a point are respectively parallel to three other non-coplanar lines through a point, the two sets of lines determine the same hyperplane or parallel hyperplanes; or if an intersecting line and plane are r espectively parallel to another intersecting line and plane, they determine the same hyperplane or parallel hyperplanes.
THEOREM 10. Two trihedral angles having their sides parallel each to each and extending in the same direction 2 from their vertices are congruent.
For the corresponding face angles are equal 3.
THEOREM 11. If a line is parallel to a hyperplane, all points of the line are at the same distance from the hyperplane; or if a plane is parallel to a hyperplane, all points of the plane are at the same distance from the hyperplane.
THEOREM 12. Two parallel hyperplanes are everywhere equidistant.
Manning, Henry Parker, Geometry of Four Dimensions, Dover Publications, Inc., New York, NY, 1956.