Unit 1: Section 0: Vocabulary

Unit 1

Section 0: Vocabulary

Welcome to the first unit, scholars! In this unit, we will be discussing the foundations of geometry. By far, the most important part of mathematics is your understanding of vocabulary! That's right; your understanding of the meanings of words is the single most important skill you can have, so thank your English teachers now for the language skills that you have that will help you pass this Geometry class.

That being said, let's dig into some vocabulary. Shall we...

Plane

A plane is a flat surface with infinite length and width, and no height.

Now, it is quite easy to determine whether a plane is 1D, 2D, or 3D. We just look at the definition, which clearly states that a plane has infinite length and also width, and no height. Well, I count: one, two dimensions. Planes are 2D.

Unfortunately, there are very few true examples of planes in the world.

Let me give you an example:

Foundations

When someone constructs a building, they need a flat surface to build on. This surface is called a foundation. This is a limited example, since foundations do not extend forever. However, since they are nearly perfectly flat by design, they serve well as examples of a plane. See below:

Figure 1: A building's foundation

When two or more points are on the same plane, they are called coplanar points. Naturally, when points are not on the same plane, they are called non-coplanar points.

Point

A point is an exact location in space.

If you ever wondered if a point is 1D, 2D, or 3D, it is none of these; a point has no dimensions. It is simply a location. Some variations on the vocab word point are:

• Endpoint
• Midpoint
• Center Point
• Intersection Point
• Vertex
• Collinear Points

Let me give you an example:

Locations on GPS

When you want to walk or drive somewhere, you use your phone or a GPS app. My old grumblings aside - about how we have no sense of direction and are therefore crippled by the very technology we invented in order to make our lives simpler (I'm rambling now) - this illustrates a handful of extremely useful Geometric objects. The first of these is a point. When you type in a desired location, your app drops some kind of marker - a pin, a dot, etc. - on that location. This is a point. Does that pin or dot have any length? Of course not. Does it have any width? Of course not. It only exists to serve as an indicator of that place's exact location in space. See below:

Figure 2: Hogwarts on Maps

Line

A line is an object that is straight and infinitely long.

Lines extend forever in both directions. A line happens to be 1D; that is to say it has one dimension. It has an infinite length, sure, but does it have any width? Of course not. Same goes for height.

When two or more points are on the same line, they are called collinear points. Naturally, if points are not on the same line, they are called non-collinear points.

Unfortunately, there are very few true examples of lines in the world.

Some variations on the vocab word line are:

• Line Segments
• Rays

Line Segment

A line segment is a part of a line that is bounded on both sides by endpoints.

Unlike lines, line segments do not extend forever. This means they have a finite length. They are still 1D, though.

Let me give you an example:

GPS... Again

When you finally get around to setting our for Hogwarts - from your home in, say, 4 Privet Drive in Bristol, your phone or computer will give you a nice little layout of all the roads and turns you will have to take. Most of these small road paths that you take are line segments. They have starting points and ending points, and they are laid out in a straight path. See below:

Figure 3: Google Maps directions from 4 Privet Drive to Hogwarts

Figure 4: A section of the directions from Figure 3

The parts of Figure 3 that are important here are the sections of road on Privet Drive and Fulford Road, as these are straight paths with clear endpoints. This makes both of these sections great examples of line segments.

Ray

A ray is a part of a line bounded on only one side by an endpoint.

No, I am not talking about the baseball team, or of undersea life, or of the famous pianist and songwriter famously portrayed by Jamie Foxx in 2004. I am speaking of a geometric object. A ray is something between a line and a segment. It has a single endpoint (like a segment) and the other end continues on forever (like a line).

Let me give you and example:

LASER Pointers

Have you ever used a LASER pointer? Have you noticed how the beam seems to go on forever? That is, if it a quality LASER, since some crappier ones are rip-offs that are made with not-so-real LASER technology with weaksauce ranges? If you have, then you've probably asked yourself, "if there were no wall here (or building, or person, or whatever), would this beam actually go on forever?" Long story short, in a perfect vacuum - like space - the answer is yes. This is a perfect example of a ray, since it has a definite point of origin (the device itself) on one end, and since it continues forever in the other direction. See below:

Figure 5: A long-range LASER pointer

Intersection

An intersection is the point (or points) that two geometric objects have in common.

You may be wondering if an intersection is 1D, 2D, or 3D, and honestly, that will depend entirely on what the objects are that are intersecting. I think this one would be easiest to describe using examples.

Let me give you an example:

Street Intersections

When we see intersections in the street, this most closely resembles an intersection of lines. This is because these two 1D objects are legitimately crossing, which means that they are going to have some point or points in common. Since they are both our equivalent of lines, and therefore would have no width or height, then their intersection will not have width or height either. It is a single point. This is a 1D intersection. See below:

Figure 6: The intersection of Mill Rd and Fire Rd is a single point

Mouldings

In houses, there are objects called mouldings. These are decorations that cover the places where a floor and wall (floor moulding) or a ceiling and wall (crown moulding) meet - hint: "where two things meet" is a very easy way to spot intersections; see also "where two things overlap". We discussed in a previous definition how a flat surface is our best example of a plane, and so the floor, wall, and ceiling are all planes - 2D objects. Where they intersect is usually some form of crack, which is why we use a moulding in the first place; that crack is in the form of a line. This is a 1D intersection. See below:

Figure 7: Moulding is used to cover the lines where walls and ceilings meet

Lighting Gels

When you work in theater production, sometimes you have to create colored light. To do this, you use color gels over your lights to give it a tint of color. A gel is a sheet of translucent paper/plastic that is tinted a color and then secured over a light. Most of these come in the primary light colors - red, green, blue - and sometimes you have to get creative to make shades like purple. This is when overlapping two color gels over a single light becomes very useful (see? called it: "overlapping" = intersection). These two gels represent planes, since they are both flat surfaces, and where they intersect is also another flat surface or plane. This is a 2D intersection. See below:

Figure 8: Assorted overlapping lighting gels

Angle

An angle is a geometric object formed by two rays or line segments sharing an endpoint.

You may be wondering if an angle is 1D, 2D, or 3D. Let's consider what it is made of: two line segments or rays that share an endpoint. Since this is the case, an angle has to be at least 1D. But, let us also consider that there is some kind of, well, angle between them, so they aren't "in line," so to speak. Since this is how angles are made, then we must conclude that angles are 2D

One important thing about an angle is that the shared endpoint is called a vertex. So, to be clear, a vertex is a specific type  of point.

Let me give you an example:

Hinges

Every pivoting door has a hinge that secures it to the wall and also enables it to move. This hinge is a great example of a real-world angle. A hinge simply connects two pivoting pieces of metal to a single rod, and that rod serves as the vertex of the angle. The pieces of metal are the two segments that make up the angle, which we will call the sides of the angle. See below:

Figure 9: A working hinge.

Attache Work

Make sure that you practice your vocabulary skills, and don't forget to take the online quiz for this section.