This page contains items of critical interest to teachers and parents teaching mathematics to children. The material here is left up for one or two months and then changed. If you want to get a past posting from here, go to the Back Fence Publishing home page, below, and send an email request through the email portal there. All material and artwork posted here is copyright of the author/illustrator, James Watt. You may cite this material or use the illustrations PROVIDED you acknowledge who wrote them or illustrated them and where they came from (www.backfencepub.com).
QUESTION du mois; "HOW MANY TYPES OF SIMPLE LINES ARE THERE?"
The word "SIMPLE" refers to forms which are composed of ‘one part or nature, only’. A simple form cannot be further disassembled into anything other than what it already is and remain 'that form'. For instance, a simple Lego brick 'is what it is'. It can be smashed with a hammer into disassembled shards of plastic, but it has ceased to be a Lego brick.
A SIMPLE LINE is, "A similar sequential series of points".
Thus, a simple line cannot change direction, since all its points are arrayed in 'similar sequence'. A line which changes direction is complex. Each change of direction is the start a new simple line. A simple line can be disassembled into simple points, but this does not change the definition of 'simple line'. The simple point is a subject separate from that of the simple line.
How many types of simple line are there? There are only two - Straight and Curved. The reason is as follows:
CHARACTER OF SIMPLE LINE.
STRAIGHT LINE; "A similar sequential series of points HAVING NO RELATION to points outside the series AB".
Illustration A shows a straight line AB and any point P "outside the series AB". The question is asked, "Is there anything about any point P which is necessary for straight line AB to exist?" The answer is "No." So a straight line has NO RELATION to any point outside its series AB. A simple line does not change direction (being of one part, only) so a straight line goes in one direction ‘as long as you want’. There is nothing you can take away from a simple straight line that will change it from being a 'simple straight line'. It either is a simple straight line - or it doesn't exist at all.
CURVED LINE; "A similar sequential series of points HAVING A RELATION to points outside the series AB."
Illustration B shows curved line AB and points P and Q. The question is asked, "Which point is inside the curve?" The answer is, "P". A curved line self-evidently shows primitive relations to arbitrary points as belonging to either 'interior or exterior' areas. The further question is asked whether there is any precise point which is necessary for curved line AB to exist? The basic geometry of any compass shows, "Yes, there is always some center point which is absolutely related in exactly the same way to each and every point on curved line AB." So a simple curved line will always HAVE A RELATION to a point outside the series AB. A simple curve ALWAYS becomes a CIRCLE (Hyperbolas, Ellipses and Ovals, etc. are complex, not simple, figures). There is nothing you can take away from a simple curve line that will change it from being a 'simple curved line'.
By the Law of the Excluded Middle (this law says when some thing either absolutely "is or isn't" then no other possibilities exist as a result) either a simple line ‘will or will not’ have a relation to some point outside the series AB. Therefore, throughout the entirety of the Universe, there are only two simple lines possible, - either a simple line is Straight or it is Curved. From this knowledge, the entirety of all mathematics is shown to be not some hodge-podge of isolated convenient formulas, etc., but is actually and inevitably the unified study of the prolific relations built up from simple straight and curved lines, only - which is pretty astonishing, don’t you think?
Something Deep and Marvelous to Ponder...
The hardest thing to prove is that some actual (physical) straight line exists ( has no relation to points outside its series AB). The easiest thing to prove is some actual line is curved. As soon as some line demonstrates a relation to points 'outside its series AB', it is identified, mathematically, as a curved line.
In 1917 it was demonstrated - as fact - that light is locally curved by gravitational lensing around stars. For light to actually curve, even once, it has to already have the pre-existing ability to potentially curve within its simple nature, otherwise - local lensing couldn’t happen. Physicists like to say light travels in a straight line, but does it really?! If light truly travels in a straight line, light could not warp around stars gravitational fields as has been clearly established since 1917. The Law of Excluded Middle then says light must be fundamentally curved and all the light in the Universe simply loops around (in gigantic minimal arcs to be sure) the Universe and will essentially "never leave the Universe".
Curved light is the fundamental unit of form in a closed Universe and such a Universe must, by form, be absolutely conserved in all its parts. If you hung around long enough in such a Universe, say, 14 billion years, you could watch the entire history of that Universe re-run as the light comes back around again. One of the distant galaxies you would see through a telescope might be the backside of our own Milky Way Galaxy, from billions of years ago. This is a far different view from the Universal models proposed on the curiously unexamined common assumption that light 'travels in a straight line, only'. Modeling light as traveling in curves would have the most profound effects on our understanding of the physics of the Universe.
If you found the above informative, it should interest you to know there is much left out of standard school math curricula which destroys or hobbles children's continuity of learning. Without the least bit of hyperbole or exaggeration I let you know, one of the most important of these omissions occurs in fundamental grade school arithmetic.
The only reason why all children don't get As and Bs in grade school Arithmetic is that critical information which shows them how to do this has been left out of all math curricula. Read about the amazing new book, ELIMINATING CARELESS ERROR at the publisher's website:
Back Fence Publishing Home Page