Models of Incidence Geometry – What Do They Look Like?

Video Series: Incidence Geometry (Tutorials with Step-by-Step Proofs)

(D4M) — Here is the video transcript for this video.

00:00
hi everyone welcome back i’m dave uh in today’s episode we’re going to talk
about models of instance geometry what do they look like
and before we get before we begin i wanted to mention that that this episode
is part of the series instance geometry tutorials with step-by-step proofs link
is below in the description so let’s get started
we’re going to begin by reviewing uh briefly what the incident axioms are so
we’re going to let point line and incident be undefined terms
we have incident axiom one where we say that two distinct points
determine a unique line and axiom two every line has at least two points on it
and a three says that three there exists
three distinct points there exists three points that are non-collinear
and so what do we mean by collinear right so three or more points are
collinear if there exists a line through all of them
and we say non-collinear if this is not true

00:01
three or more lines are concurrent if they’re just a point incident with all
of them and then non-concurrent is the negation
and then we have the uh parallel uh definition here so
our definition of parallel means they’re distinct lines and no point is incident
with them so in the previous episode we talked more more about parallelism
uh generally speaking though uh in the first couple of episodes of the series
we went episode by episode and we proved each one of these statements here
in both column format and proof and paragraph proof format so i recommend
checking out all those episodes then um i decided you know hey we’re
pretty good at writing proofs now let’s just prove all these results here and so
we did that in the previous episode as well we went through each of these
proofs here step by step and went through all of them
all right and so another axiom in our previous episode we talked about

00:02
parallelism and we talked about these three statements here we call this
statement right here the elliptic parallel property
and this statement right here the euclidean parallel property and the
statement right here the hyperbolic parallel property and they all have to
do with if you start with an arbitrary line and a point not on the line and
they ask the question how many parallel lines go through that point
but are parallel to l and so this says no lines and this says
a unique line exactly one and this says more than one it always happens that
there’s more than one so um i just wanted to mention that
these are not new axioms for incidents geometry rather they’re just additional
statements that may or may not be satisfied
by a particular model for the incident’s
geometry and so that’s what i’m going to
talk about today are models i’m going to begin our discussion and then we’ll do
some more episodes over additional models for our instance geometry so let’s be

00:03
very clear what i mean by incidence geometry so any collection of points and
lines that satisfy the incident axioms right we have the incident relation
and all three of those terms are defined
by a1 a2 and a3 so we’re not adding this as an axiom but we’re considering it as
a question and whether or not we can or whether or not we even should right so
anyways if we have a1 a2 and a3 this is what i’m going to call incidence
geometry for now now keep in mind also that we’re just working in the plane if
you want to work in higher dimensions well then we’ll need some more axioms
for that so for right now this is what i’m calling incidence geometry
and then we’ll look at higher dimensions later
all right so right now we’ll just have a1 a2 and a3 and these are interesting
statements and we’re going to see if these statements hold and by looking at
various models all right so here’s our first model so
model is an example and i’m going to call this model three-point geometry

00:04
and what is going to be a point and what is going to be a line and what does it
mean to say incident so i need to explain all of that so we’re going to say that
in three point geometry we’re going to mean a point to be a symbol a b or c
and a line to consist of exactly two of these points um and so we have this um
this table right here which kind of helps us understand the geometry right
here so we have three points a b and c and we have exactly three lines so this
is line one which is made up of points a and b and nothing else and line two a
and c and nothing else and line three b and c and nothing else and so what we
can see is that each point is incident with two lines
so for example a is incident with these two lines and b is incident with these
two lines and c is incident with these two lines so we can see that number one
is true we can also see there’s no parallel lines for example these two are

00:05
not parallel because c is on both of them
these two are not parallel because a is on both of them and l1 and l3 are not
parallel because b is on them so there’s
no parallel lines so every time you take a line and a point not on it
the elliptic parallel property holds because there’s no lines that are
parallel through it so what this shows is that the elliptic
parallel property can hold in an instance geometry well we’re going to
see some examples of some other models for example we do four-point geometry
and five-point geometry we’ll see that some of these parallel properties
some of them holds in other models another one might hold
so we’ll look at that here in a couple couple of episodes so this is the first
example of our first model here and so what we want to check though is
in order for us to say a model we can’t just you know put it all down and say it
is we have to actually check every axiom right so for every point p and every
point q not equal to p in other words if

00:06
you pick any two points there’s only one line going through it so there’s
six possibilities to check a and b and here’s the unique line that goes to
a and b a and c and so here’s the unique line and then
b and c and then here’s the unique line and then what if we switch the ordering
right b and a well it does doesn’t matter b and a is the same thing if we
have a and b either way it’s the same line
so all possibilities considered we can see that this axiom holds
and this is just a brute force check every possible combination and we see
that axiom a1 holds when we say here for every
right we went through and checked every p and every q all the possibilities and
we can see that a1 holds uh a two holes does doesn’t that true
for every line so we’ve got to check all
the lines there’s only three lines so we can check all three does each line have

00:07
at least two points on it and the answer
is yes you can just tell that by looking so a2 holds what about a3
uh there exists three distinct points right so three-point geometry is going
to be the minimum because of a3 right there exists three distinct points
and no line goes through all three of them right so there’s no line that goes
through all three of them l1 doesn’t go through all three of them l2 doesn’t go
through all three of them and l3 doesn’t go through all three of them so axiom
one two and three a one eight two and three hold so we can say this is a model
this is an example and here’s two properties that we talked about that
stand out in this model this isn’t two properties that’s gonna hold in every
model as we look at different models we can make additional statements but in
order for us to first call it a model we have to check that all three of these
actually hold all right so now let’s take a step back
and look at the axiomatic method a little bit more detail

00:08
in the first episode of the series where i talked about what instance geometry
was i touched on some of these terms but now i want to go through this a little
bit better or not better but necessarily that you’ve seen what theorems are
you’ve seen what a model is you’ve seen you know a lot of this stuff here now
for several episodes you’re you’re becoming more familiar with it so let me
just go through here and you know rehash or or you know make sure
that you’re okay with all this stuff right here so we have undefined terms
must contain technical terms that are deliberately chosen as undefined
and subject to the interpretation of the
reader and this is what models are going to give us they’re going to give us an
interpretation and so you know because in the
three-point geometry nothing in the axiom says that you can only have three
points that’s what three-point geometry says you only have three points right so
that’s an interpretation the three-point geometry is

00:09
all right um so we also have defined terms such as collinear and concurrent
lines and so we also have defined terms and then we have the set of axioms
system uh system axioms of the system are set as statements
dealing with the defined and undefined terms and remain unproven
so the word postulate is another word for axiom there
all right and then all other statements are called theorems
now the more mature or more advanced your system becomes then you’ll start
distinguishing between limits and theorems and propositions and facts and
things like that i talked about a little bit about that in a previous episode
all right properties of an ectomatic system there are some definite
properties that we want to talk about so um it’s important for the system to
be consistent so inconsistent means you can derive a contradiction
and you don’t want to be able to drive a

00:10
contradiction if your system derives the contradiction then it’s basically
useless it’s not that it’s not undefined maybe your system makes sense
but it’s just not going to be used for anything useful because your system will
say anything and everything so an action system is called consistent
provided a theorem any theorem which can be proven
can never logically contradict any of the axioms or previously proven theorems
so that means it’s consistent another property is is called
independent so an axiom is independent if it cannot be proven from the other
axioms so if we want to actually add one of the parallel postulates
the the elliptic euclidean or hyperbolic if we want to add another axiom to our
system um we would be we would be able to show
that it’s independent by looking at different models um and so
that’s that’s another reason why models can be useful you can use models to help

00:11
you understand if a set of axioms is independent or not from
one statement is independent from the rest of the statements
so generally speaking you’d like to have a
set of axioms that are independent for each other
but it’s it’s not necessarily a necessary requirement
uh sometimes you may have uh two alternative motives for giving your axioms
maybe they’re just easier to get started with whatever
maybe some seem more intuitive so this one is not as an important
property as consistency consistency is a must-have property
independent set of axioms is is useful extremely useful especially for people
who are building and working with axiomatic systems
an axiom set must also be complete so this this word is used a lot of
different ways in mathematics and so the the way i’m

00:12
going to mean it here is you must have enough axiom to prove every theorem
that can possibly be stated using the terms and axioms of the system
that can be proven either true or false in other words if you can write out a
theorem and it it has a true or false value you
need to have enough axioms to actually get to whether or not it is true or false
all right and so um let’s ask the question is incident at geometry
consistent right so instant geometry remember we we said that was axiom a1 a2
and a3 so the immediate question is is it consistent
so let’s look back at our model we can use models to help us answer this
question so three-point geometry we have points a b and c
and we have these two properties that we talked about and here was the geometry
given to us in this table right here so the question is is it consistent though
and here’s how we’re going to answer that question so to answer the question

00:13
we’re going to say maybe it’s inconsistent in fact let’s
just say it’s inconsistent the incidence geometry is inconsistent
in other words if you look at a1 a2 and a3 and you just look at those axioms you
can derive a contradiction from them well
then that would mean that you’d be able to derive any statement at all from the
system and so we’ll be able to have the statement right here if points a and b
are distinct then a and b are the same points so if your system is inconsistent
you’d be able to derive anything from it and you would be able to derive this
statement right here from it if a and b are different then a and b
are the same point right that’s obviously a contradiction
and if we look at this geometry right here it doesn’t actually hold does it
so if this was a theorem in our geometry
then it would have to manifest itself in this model right here and it doesn’t i

00:14
have three distinct points and none of them are the same so
this uh assumption right here that it’s inconsistent doesn’t work out
because if such a proof existed then it would hold in any model of it and it
doesn’t hold in this model so even just coming up with one model of a geometry
is valuable and could be valuable information in particular a proof of
such statement would have to hold in three-point geometry and it doesn’t
so there’s exactly three distinct points there’s no way we’d be able to prove
this statement right here so we can conclude that incidence
geometry is consistent theory so um again you know if it’s
inconsistent then anything can be proven and it’s a useless system
so this model right here gives us an example
of how to show that something uh how or how to get evidence that something is
consistent right here all right so this is um exactly what i
just said right there so all right so now let’s uh talk about

00:15
more models yeah let’s right here click right here and we’ll talk about 4.5
point and we’ll do some more models fianna’s geometry
click there and uh i’ll see you in that episode

About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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