Parallel Lines and Planes
Parallel lines are lines in the same plane that never meet. Never? That's what Euclid's fifth postulate said. And Euclid's geometry is the same geometry that has been taught in schools for over two centuries. That is why many textbooks and courses are called "Euclidian Geometry".
If we accept this postulate (And why wouldn't we? We have accepted all the others!) then we can, in this chapter, make some important discoveries about parallel lines and the angles related to them.
So far in geometry, we have mostly talked about what are called "plane figures", that is "flat" figures that lie in a plane, that we can draw on a piece of paper. But there are other figures - 3-dimensional ones, like cubes and pyramids. The world around us is, of course, 3-dimensional: objects have length, width, and height.
In this photograph, you see a 3-dimensional object that should be familiar to you, a building:
This beautiful house was designed by Frank Lloyd Wright, perhaps America's most well-known architect. The house is called Falling Water and was built over a waterfall, in Pennsylvania. If you would like to learn more about Frank Lloyd Wright, and see more of his architecture, click on the link below to visit one of many websites devoted to his work.
The sketch below is an attempt to draw Falling Water, and portray it as a 3-dimensional object by drawing it on a 2-dimensional surface.
There are a number of different ways to draw pictures that appear to be 3-dimensional. One method is perspective drawing. The drawing above was drawn in perspective. Another kind of 3-D drawing is called Isometric. A third method of 3-D drawing is called Oblique. The pictures below illustrate these 3 kinds of 3-D drawings:
If you click on the link below, you will find an isometric gridwhich you may use to do an isometric drawing. Use a sheet of tracing paper placed over the grid paper, and then you will be able to use the grid again later to do another drawing.
A more detailed drawing of a Perspective is drawn below, showing the method of laying out a drawing in what is called Two Point Perspective.
There are 3 things that are always true in this type of perspective: 1) Lines that are vertical on the real building are drawn vertical on the drawing of the building. 2) Lines that are horizontal on the left side of the real building are drawn to a point on the left of the drawing of the building, called the Left Vanishing Point . 3) Lines that are horizontal on the right side of the real building are drawn to a point on the right of the drawing of the building, called the Right Vanishing Point.
One of the most fascinating aspects of Mathematics is that there exist statements that are both true and false. Perhaps the most famous of these is Euclid's controversial fifth postulate. In the words we use in modern geometry, this postulate said:
"Through a given point only one line may be drawn parallel to a given line." In the drawing below, we see line m, point P, and one line through P parallel to line m. This is the only parallel that can be drawn.
It certainly seems reasonable enough. Interestingly, throughout history, mathematicians have tried, in vain, to prove or disprove it. "It seems that Euclid himself did not entirely trust the postulate, for he avoided using it as long as he could in his great work, The Elements, by proving his first 28 propositions without it.
Mathematicians have questioned the validity of this postulate for centuries. Some disagreed and declared that there could be many lines parallel to a given line, others said no parallels could exist.Today, there are two main classes of geometry: Euclidian and Non-Euclidian. Non-Euclidean geometry consists of both the Bolyai-Lobaschevsky and Reimann models. It has been sufficiently proven that Euclid's fifth postulate can be both true and false. By assuming it true, one can generate the geometrical world known as Euclidian Geometry in which no contradiction has arisen. Likewise, by assuming a different theory of parallels, one can generate another, quite separate geometrical world in which no contradiction has been found."
(From a web page by Oxford Web Productions.) To read the whole paper, click on the link below:
It may seem strange that anyone would think that you could not draw a line parallel to our line m through our point P. How could anyone argue with this? Well, it depends on your meaning of line, and in what sort of space you are drawing this. What about on a curved surface? Why would we consider a curved surface? . . . well, because we happen to live on a curved surface, the planet Earth! If we define line to be only lines of longitude (the ones that run through the North and South poles) then none of these lines are ever parallel because they all intersect! So the Fifth Postulate would have to say: "Through a given point, no lines can be drawn parallel to a given line." Why would we find this definition of line useful? Because that is how we navigate the Earth, by ship or by plane! This kind of geometry is called Spherical Geometry, or Riemannian Geometry.
This image came from an interesting website with information and images related to the Earth. If you are interested in learning more about this, click on the link below:
And if you were to define line as some other sort of curve, then other geometries might follow from that. Image a curve called the Hyperbolic Paraboloid (the shape of a saddle, or of a Pringles potato chip!) "Lines" could easily be drawn on this saddle shape in such a way that they did not intersect! Many lines could be drawn "parallel" to a given line (parallel in the sense that they do not intersect). But this seems pretty "far out" and of no use, except as an intellectual exercise. Where would we encounter such a curved shape? Well, Einstein said that space is curved, and perhaps this geometry makes more sense when dealing with space travel, or in astronomy! This kind of geometry is called Hyperbolic Geometry or Lobaschevskian Geometry.
If you are interested in learning more about this curve, and other 3-D curves, click on the link below:
In summary, there are 2 kinds of geometry: plane geometry, which is most of what we are studing in Connecting Geometry, and 3-dimensional geometry. In the following project, you will be learning more about 3-dimensional geometry, and 3-dimensional drawing.
Learn more about 3-D drawing by going to the Math Forum website and reading all the pages at this link: http://www.forum.swarthmore.edu/workshops/sum98/participants/sanders
Open a new GSP file. Draw a horizontal line across the middle of the page. Label the 2 endpoints of the line VPL and VPR. Use these vanishing points to draw a simple building of your own design, in perspective.
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