Welcome to the home of Levelology, we seek a balance and understanding.
Do you question the little things you take for granted?
Find Reality and Start to Lose the Model
The next twelve steps may cause you to reconsider the reality you've been given, you may find the perception of what you think you know to be at odds with what you've been taught. There's hidden depths within what we call 'Advanced Simplicity'
Some of these facts may cause you pain, please read the warning below.
Straight lines don't bend or curve
Straight line is Right
A RIGHT line is that which lies evenly between its extreme points, that is without curvature in any direction being the shortest path between those 2 points, never deviating from that path.
This is not a concept but a fact of reality and basic logic.
Euclid's Elements Book 1 Definition 5
The Plane (superficies)
A superficies (plane surface) is that which has ONLY length and breadth
Those two dimensions - Length and breadth make this a 2D conceptual surface on which Plane geometry is based.
This 2D plane obviously does not enter into 3rd dimension.
Euclid's Elements Book 1 Definition 10
Perpendicular
When a straight line standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called perpendicular to it.
As stated in Axiom 11, all right angles are equal therefore the lines must be STRAIGHT
Euclid's Elements Book 11 Definition 3
Perpendicular to a Plane
A straight line is perpendicular, or at right angles, to a plane, when it makes right angles with every straight line meeting it in that plane.
Book 11 Proposition 6 states; if two straight lines be at right angles to the same plane then they shall be parallel to one another.
Let's get real
Vertical
At right angles to the horizon; perpendicular; upright.
In mathematics it is a plane, position or straight line.
Not just a straight line
Horizontal Plane
Parallel to or in the plane of the horizon.
At right angles to a vertical line.
In mathematics either measured or contained in a plane parallel to the plane of the horizon.
Cartesian axis
Horizontal and Vertical
Horizontal and Vertical geometrically are perpendicular, with Horizontal being the plane and vertical being perpendicular to it.
They both have a common factor being the horizon, which they are either parallel or perpendicular to.
Cartesian axis x,y,z
Reference frames Horizontal and Vertical
In reality horizontal and vertical are the reference frames we use as our baseline, any inclined plane will be measured as an angle to one of these reference frames.
Perpendicle
Plumb Bobs and Perpendicles
A perpendicle is ANY thing hanging down by a strait line.
A Plumb Bob is a tool for establishing a vertical reference, this is used in reality daily by many people.
A vertical line is a straight line.
Level is a right line on a horizontal plane
Level
Level is a straight line on a horizontal plane.
A level is a tool for establishing a horizontal plane of reference from which vertical distances can be measured.
An instrument for determining whether a surface is evenly horizontal.
From Latin libella "a balance, level".
Perpendicularity
Its all right
The relationship between reality and geometry becomes apparent with the right line relationship between the 2 frames of reference used in everyday life.
Level and Plumb are perpendicular.
Vertical and horizontal are perpendicular.
This is NOT a coincidence - Geometry is reality.
Here's the reality link again
Water level (tool)
A water level is any device utilising the surface of liquid water to establish a horizontal plane of reference.
The water level is lower tech than the more modern levels, but it can be more accurate over long distances, and works without a sightline, such as around corners.
2 ends = 2 points at an even altitude.
Euclid's Elements Book 1 Proposition 30.
T-Shirt Logic
All horizontal planes are parallel.
Book 1, Proposition 30.
Straight lines which are parallel to the same straight line are parallel to each other.
Book 11, Proposition 9
2 right lines which are parallel to each other.....
Once again the link between reality and geometry becomes clear, you already know it to be true.