Given a dimensionless point, we could define an infinite number of points at different locations throughout the universe.
Given 1 stationary point A and varying another point B, we could define an infinite number of 1-dimensional lines intersecting A. There would exist lines that did not intersect A. Although these lines could be skew, we will call them parallel to the lines intersecting A or B.
Given 2 stationary points A and B, and varying another point C, we could define an infinite number of 2-dimensional planes intersecting A and B. There would exist planes known as parallel planes which did not include A, B, or C.
Given 3 stationary points, A, B, and C, and varying another point D, we could theoretically define an infinite number of spaces intersecting A, B, and C. There would also supposedly exist parallel spaces not containing points A, B, C. D.
Also, the intersection of any two intersecting lines is a point. The intersecting of any two intersecting planes is a line. The logical continuation implies that the intersecting of any two intersecting spaces is a plane.
Summary: Given n stationary points, and varying another point, we could define an infinite number of regions with n+1 dimensions. For any region, there are theoretically parallel regions with the same number of dimensions, which do not intersect the first region. The intersection of any two intersecting n-dimensional regions is a n-1 dimensional region.
Implications: Given the ability to move in a new direction, (such as an ant existing in a plane suddenly gaining flight), we could either 1) Move from our current n space to a n+1 space (i.e. higher dimensions), 2) Move from our current n space to an intersecting n region (i.e. an alternate dimension), or 3) Move from our n-space to a parallel n-region (i.e. a parallel universe). Note that in the cases of 2 and 3, we would not need to retain our newfound ability to move in a new direction.
And there you have it. That's my take on the science of science fiction.