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The World of Quadroid:  Four-Dimensional Euclidean Geometry

by Bruce G. Marcot, 1969

 Background:  At age 16, I found myself about 1.5 years advanced in high school, and was taking higher-grade courses in mathematics.  It was in one such course, pre-calculus algebra, that I found myself so bored by the coursework that I took it upon myself to rewrite a portion of the textbook, expanding it to a 4-dimensional geometry, inventing the postulates and theorems as I went. 

            This is clearly a reinvention of what has been developed more fully by many others, I am sure, but at the time I had no knowledge of that.  I termed 4-dimensional space “quadroid,” from the Latin quadri- meaning four and –oid meaning form. 

            What struck me is how logical and consistent were the concepts of parallel spaces, skewed spaces, and rather paranormal-sounding ideas that arose from this treatise.   

            Most of my notes from that period have been lost, and this is the only portion that remains. 

 
 

1.  Quadroid (4-dimensional space) geometry assumptions.

Postulates:

(1) There is a set of points called quadroid.  
            Certain subsets of quadroid are called spaces, planes, lines, and points.  Quadroid contains at least five points not in any one space.

(2) There is exactly one space containing any 4 noncoplaner points.

(3) Each space satisfies postulates for quadroid geometry.  
            In particular, the plane joining 3 points of space lies in that space.

(4) Each space separates quadroid.  
            This means that the set of points not in the space consists of two subsets on either side of the space, called half-quadroids, with the following properties:  (a) If two parallel lines are in the same subset, the plane joining them does not intersect the space.  (b) If two parallel lines are in different subsets, the plane joining them does intersect the space.

 Remarks:

 The first postulate asserts that no space fills up quadroid.  We have a geometry of more than three dimensions.

 The second postulate gives a simple definition of determining a definite space.

 The third postulate makes our spaces like those familiar from space (3-dimensional) geometry; all spaces are alike.

 The fourth postulate makes quadroid four-dimensional.

 Theorems:

 1-1.  If a plane, p, does not lie in a space, π, and contains at least one line of π, then p and π have exactly one line in common.

 1-2.  There is a unique space containing a given plane and a line not on the plane.

 1-3.  There is a unique space containing two distinct intersecting planes.

 1-4.  There are infinitely many spaces containing a given plane. 

 1-5.  If two distinct spaces have at least one line in common, they have exactly one plane in common.

 

  2.  Congruence, parallelism, and perpendicularity.

 Definitions:

(1) A plane, p, and a space, s, are perpendicular (p ┴ s) if p intersects s at a line and p is perpendicular to every plane of s through that line.

(2) Two planes are parallel if they are in the same space (are cospacer) and do not intersect.

(3) Two planes which are not cospacer are skew.

(4) Two spaces are parallel if they do not intersect.

(5) A plane and a space are parallel if they do not intersect.

Theorems:

2-1.  If a plane, p, is perpendicular to two intersecting planes, p1 and p2, then the given plane is perpendicular to the space containing p1 and p2.

2-2.  There is exactly one space perpendicular to a plane p at a line l of the plane.

2-3.  There is a unique space perpendicular to a plane p and containing a line l not on p.

2-4.  All the perpendiculars to a given plane p through a line l lie in the space perpendicular to p at l.

2-5.  If two planes are perpendicular to the same space, they are parallel.

2-6.  Through a line, l, not in a space, s, there is exactly one space parallel to s.

2-7.  Two distinct spaces perpendicular to the same plane are parallel.

2-8.  If a plane, p, is parallel to a space, s’, and space s” contains p and intersects s’, then p is parallel to the plane of intersection of s’ and s”.

2-9.  If two distinct planes, p1 and p2, are parallel to a plane p, then p1 and p2 are parallel.

 

 3.  Spacer angles and perpendicular spaces

One of the basic assumptions of space (3-dimensional) geometry is that every plane, p, in a space, s, separates the points of s other than the points of p into two disjoint sets.  Each of the two sets is called a half-space.

The same way that dihedral angles are formed (two intersecting planes at a line), spacer angles are formed:  two intersecting spaces at a plane form a certain angle between those spaces. 

Theorems:

3-1.  Any two spacer angles of the same spacer angle are congruent.

3-2.  Two angles whose sides are parallel and in the same direction are congruent.

 

4.  Quadroid coordinization

Space coordinization utilized three dimensions, three axes coming together at a single point, all forming right angles to each other.

To coordinize quadroid (four-dimensional space), a fourth dimension must be shown – a fourth line intersecting at this point, such that all 4 lines are perpendicular.  This cannot be shown in three dimensional space, as the fourth line pierces into the fourth dimension; however, perspective views projected into three dimensions can be visualized.

Space coordinization consists of the x, y, and z axes.  The fourth axis will be labeled q. 

Points are therefore labeled as p = (x,y,z,q).

 

5.  Distance formula

For any two points in quadroid:

            p1 = (x1, y2, z1, q1)

            p2 = (x2, y2, z2, q2)

the distance formula =

     

 

6.  Lines

 Parametric equations and direction cosines:

 

Parametric equations are:

 

where d is a parameter and can be any real number.

 

Theorems

6-1.  If C1, C2, C3, and C4 are direction cosines of a line, then:

  

6-2.  If a, b, c, and d are real numbers and if    then there is at least one line whose x-, y-, z-, and q-direction cosines are a, b, c, and d, respectively.

 

7.  Spaces

 

If A, B, C, and D are real numbers, not all zero, then the set of points (x,y,z,q) such that
            Ax + By + Cz + Dq + E = 0
is a space.  Every space, conversely has an equation of this same form.

 

Intercepts and traces of spaces:

    The space Ax + By + Cz + Dq + E = 0 has the intercepts:

    (-E/A, 0, 0, 0), (0, -E/B, 0, 0), (0, 0, -E/C, 0), (0, 0, 0, -E/D).

 

Traces of the plane is given by ......  

[here, the remnants of my notes end, but I had gone on.  Way on.  
Maybe some day I'll reconstruct the rest of this.]

 

© Bruce G. Marcot

 

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