#Axioms of incidence

21 messages · Page 1 of 1 (latest)

near galleon
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Hello, these are by my prof. and formulated in a weird way so I need a bit more explanation on them:

1: Each line contains at least two points.

2: There is at least one line containing two (given) points.

3: There is at most one right containing two (given) various points.

4: Every plane contains at least one triple of non-collinear points.

5: There is at least one plane containing three (given) points.

6: There is at most one plane containing three (given) non-collinear points.

7: If two different points of a straight line belong to a given plane, then all points of that line belong to that plane.

8: If two different planes have at least one point in common, then they have at least one other point in common.

9: There are four non-coplanar points.

Okay so, 1. is trivial, 2. I am not sure about, it is true if A = B for example but then it wouldn't follow the definition of a line and for A != B 'least' doesn't make sense as it is then literally 3.
3. trivial, 4. trivial, 5. again, same like 2. maybe if A, B and C are collinear, then it forms a line that can be a part of multiple different planes. 6. trivially true for euclidean geometry, 7. Similar to 1. but involves a plane, 8. this is the one I didn't understand quite, it's similar to 5. (for collinear points making a line) and 9. also isn't quite understandable when it can be defined with 3 different points in 3D space on 3 different planes. I don't see the point of the 4th point.

Somebody please explain these or I should just learn them as is and not question them (they're axioms after all I am a silly mf)

marsh anchorBOT
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heavy ravine
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are you trying to prove the axioms?

sturdy orbit
loud crescent
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son

near galleon
heavy ravine
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for 8: planes either coincide (intersection: plane), intersect at a line (intersection: line) or are parallel (intersection: none)

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it is impossible for them to meet at a point only

near galleon
# heavy ravine for 8: planes either coincide (intersection: plane), intersect at a line (inters...

coincide intersection doesnt make sense (I assume I am doing euclidean geometry so shouldnt then these planes be subset of one another?).

Now, point only is my problem. Let's say we have 2 square planes. If one is laying horizontally, and the other one is vertical and rotated by 45 degrees, then it has a sharp point, so if we lower it down just enough to include only the vertex of the square (vertical oriented plane). So, then they meet in only 1 point no?

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Or my assumption that theyre finite planes is wrong

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So if both planes are infinite then I guess both of the statements I am questioning are true

heavy ravine
sweet tendon
sturdy orbit
near galleon
near galleon
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I start questioning too much stuff although yeah plane is infinite by definition and so is a line

plush drumBOT
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@near galleon

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near galleon
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+close

plush drumBOT
# near galleon +close
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