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hi everyone welcome back today’s uh episode models of finite incidence

planes we’re going to be looking at young’s geometry

and this episode is part of the series incidence geometry tutorials with

step-by-step proofs uh in this episode we’re going to continue giving some

models and so let’s briefly review what incidence geometry is so we’re going to

be looking at these three actions right here and we’re going to be calling them

an incidence geometry so for every two points there’s a unique line going

through them for every line there exists two points on it at least

and there exists three non-collinear points so those are our three axioms

anything that holds for these three axioms we’re going to call an instance

geometry and we’re going to continue looking at some models now in previous

episodes we proved these 11 theorems hold for any incidence geometry

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we did these step by step in previous episodes and then we talked about three

additional statements that may or may not hold in fact we get we gave a model

three-point geometry where the elliptic property held

and the four-point geometry where the euclidean property held

and in five point geometry these were done in previous episodes

um and then in our last episode we talked about finals geometry and the

ellipt and we showed that the elliptic parallel property was true or we

discussed uh that is true all right and so today let’s

talk about young’s geometry what is young’s geometry so first of all it’s a

model for incidence geometry that means axioms a1 a2 and a3 hold

and what we’re gonna have is nine points or nine symbols each one of these are

going to represent points in our in our geometry so uh a through i

and we’re gonna have the meaning of incidents is going to be

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any of these points as uh on a line and it consists of one point and two

other points and so here’s our um you know actual

concrete lines right here right they’re just spelled out for us all the way

and so we can understand the uh statements of our geometry much more

clearly and we can see what our points and we can see what our lines so we have

nine symbols and we have 12 lines and in order to call this an incidence

geometry we need to show that axiom a1 a2 and a3 hold

so let’s discuss this first um so i always think a2 is the easiest

um because what it means is for every uh for every line there’s at least two

points on it right so as we can see here every line has three exactly three

points on it so we definitely got a2 you know each line has at least two points

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on it and you can just see that all right so just by observation

uh what about a3 so a3 says there exists three non-collinear points uh a b and d

what about them are they non-collinear let’s look for a line

and see if a b and d are on one line uh so a b and d anyone

and so as we check here for a b and d all in one line and we can see we don’t

have them on one line so they’re non clean year

now could there be other points that are non-clinical absolutely i’m sure it

would be easier to find three more but axiom e3 just says you have one pair uh

or one triple right these three are non-collinear so eight axiom a3 is done

um you know notice the difference between a2 and a3 a2 i had to check

every line had at least two points a3 is in existence it just says there exists

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three you can name any three that you want and they’re non-linear

all right so now what about a1 here so a1 says

for every two points there’s only one line going through it

um and so when i have these two points here a and b uh a and b can no longer be

on any two any other line right because they’re on

this this line here so if i check any other line a and b better not be on it

so if i check all the other lines there’s no a and b on it

right this one’s got an a on it and this one’s got a b on it but a and b are on

both on this line and there’s no other line where a and b on it well that’s one

case what about a and c here’s a line with a and c on it are

there any other lines with a and c on it no

right so there’s a unique line so i’ll write the unique line right here it’s l1

and l1 and what about a and d so axiom a1 says we have to go through every two

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every two distinct points every pair and find one and only one line that goes

through them what about a and d well we have l2 right here goes to a and d are

there any other lines that go through a and d no so it’s just l2

and so then i’ll check a and e and then a and f and

all the way to i right so a and g a and a and h and in each case we will check

how many lines go through it this should only be one let’s check a and i

how many lines go through a and i so then then here’s one line that goes

through a and i it’s l4 are there any other lines that go through a and i

always we check here none of them do so a1 is satisfied for this case and if

we check every single case then we will see that a1 holds all right

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so you got to go through all the pairs there and this is a brute force technique

of just by looking at a concrete geometry and checking every single one

here um right and so then once we’ve gone through all the a’s and checked out

all the cases we can start with b and we can check bc bd be bf

and we can check all those cases and make sure there’s only one line that

goes through uh both of those um points there and then we’ll and then

we’ll uh finish the b so then we’ll do the c’s and d’s and e’s and f and all

the way and you have to check every case so i i’m assuming that you’re going to

go and check all the cases and make sure that this is right here

i’ve already checked all the cases i’m not going to put all the cases inside of

the episode here all right so now we can say that this is a

incidence geometry young’s geometry is an incidence geometry and we got the

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point so we got the lines and we verified the axioms a1

we verified case by case by case you have to verify each and every case if it

fails even in one case then it’s not an incidence geometry all right so

we checked all cases here i checked all cases here

all right so now what can we say that’s additional

to young’s geometry not only is it an incidence geometry actually there’s a

lot of incidence geometries so some of the geometries um had

different properties for them for example three point four point five

point and fiona’s geometry right so they had different parallel properties right

so each one each geometry that you look at

who knows which parallel property might hold true so and young’s geometry here

so the first thing is the first property is each point is incident with exactly

four lines right so this is each point right so

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what are the cases that we have to check

to see that this statement is true so we have to check point a b c d e f g

h and i and for each one of these we have to

come through and show exactly four lines so let’s count what lines um are is are

what lines is a on right so one two three four so l1 through l4 1 l2 l3 el4

uh is a on any more of them just physically check just observe right a’s

not on any more of them so it’s exactly 4. what about b is b on any of them um

one let’s write it down a one and then l five and then l six l seven

and then l8 no no no uh and we got our four lines right there

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we got our four lines right there so we have to check all the way through to i

let’s check i all right i’m saying this right here

statement is saying exactly four lines so what are the four lines that i is on

one two three and four and so we just write them down right

here now four this is the evidence right here one l6

and then usually evidence doesn’t add up to a proof unless you’ve checked all

cases each and every single case and so i’m going to write dot dot here

because i have confidence in you that you’ll sit down and do this l4 l6

l8 and l12 pause the video and fill in these steps right here for us

all right the next uh thing that i noticed is that

for every two distinct points there’s exactly five lines that are not incident

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with these points so what are the cases we have to check

for every two distinct points right so we’re gonna have to check the

case a and b two distinct points right and i’m saying there’s exactly five

lines that don’t go through either of these so let’s see here

no this one goes to a a a b b b and here

we go one two three four five these five

lines right here do not go through a nor b so l eight through l

so exactly five lines right there one two three four five so what about ac

is there exactly five lines that do not go through these points let’s check a

and c it’s got a a uh here’s one of them so let’s write down here l5 and

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a and c i’m looking at so l6 and l7 let’s write those down l6 l7 um l8’s got c

c c and then l 11 and l 12. and there’s five lines right there that

do not go through a and c so you have to check every single case here

all the way to f and g what are the five lines

that do not go through f and g right so l1 l2 goes to g um l3

l3 and then l4 goes through f and l5 and then l6 and that goes to g and then l8

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and then goes g f f g and so there we go those are the five

lines there one three five six and eight those are the three fives uh three the

five lines for that case but we gotta check every case in here to make sure

that this line that this statement 2 is true alternatively

you could try to prove a1 a2 and a3 and try to prove step number two

however you will fail there are other incidence geometries

that satisfy a1 a2 a3 in fact every incidence geometry has to satisfy these

but this property right here does not hold for every incidence geometry

so you cannot prove statement two from these uh statements right here you

would have to make some additional axioms and you would have to actually

verify those axioms with these points here and these lines

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all right step three the euclidean parallel property holds so what that says is

through every line any line at all in any point not on the line

there has to be one unique line that goes through p that’s parallel and so

in order to check the euclidean parallel property there are a lot of cases

because you got to check every line and for each of those lines for all 12 lines

you also have to check every point that’s not on the line so let’s make

sure we have an understanding of what all the cases would be

so again we have to check every line so we would start off with say check line

one check line one and we need a point not on the line now

line one has three points but how many points are in our geometry we got nine

so that means we’ve already got three so that means we need

to check six cases one two three four five six six cases there so a case for d

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and e and f right and then g and then h and then line one and then i so

i’m checking line one and a point not on it

and as you can see for line one there’s going to be six cases we have to check

so six cases and then how many cases do we have total

right because we got 12 lines so we’re going to have 12 times six cases total

cases so as you can see we’re starting to get to the point where

uh this is not necessarily fun to go through all these brute force cases and

check and so um you know imagine how we check

things when we have infinitely many points right we’re not going to do

things by brute force we’re going to have other methods for doing this and

i’m going to show you those methods in an upcoming episode of upcoming episodes

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so this euclidean parallel property you have to go through and test twelve times

six total cases and we will find out uh that it works

let’s just look at one case for examples these you can go through all of them on

your own so l one and d right l1 is a line and d is not on it and what we’re

going to say is the euclidean parallel property says is so now this is l1 and

this is d and there has to be one line through d

exactly one line that’s what euclidean parallel property says one line through

d that’s parallel to l one so let’s check uh here’s line l1

so to be parallel means it cannot have an a b and c on it it has to go through d

so this goes through d but it has an a on it so that doesn’t count

uh that doesn’t go through d doesn’t go through d

all right here this one goes through d but it has a point b on it so that one

doesn’t count uh are there any other lines that go through d

here’s a line that goes through d right here

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and it has a c in common so that one doesn’t count

um here’s a line here d that goes through d

and it is parallel so this would be the line l1 through here or sorry l11

and it’s the only one l12 doesn’t go through d right so you

know this case right here works because of l1 so i’ll just draw an arrow l1

right here so l1 is the unique line that passes through d that’s parallel to l1

and we go through all of these cases and we find the unique line that passes

through the point not on the line and then we’ll verify the euclidean

parallel property holds for young’s geometry

all right well that’s it for uh this episode i look forward to seeing you in

the next one click right here for the next episode i’ll see you there