Intersection theorem

Key takeaways

Intersection theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Intersection theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Intersection theorem from memory before moving on to harder problems.

Reference excerpt

In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects A and B (for instance, a point and a line). The "theorem" states that, whenever a set of objects satisfies the incidences (i.e. can be identified with the objects of the incidence structure in such a way that incidence is preserved), then the objects A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't. For example, Desargues' theorem can be stated using the following incidence structure:

Points: { A , B , C , a , b , c , P , Q , R , O } {\displaystyle \{A,B,C,a,b,c,P,Q,R,O\}}

Lines: { A B , A C , B C , a b , a c , b c , A a , B b , C c , P Q } {\displaystyle \{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\}}

Incidences (in addition to obvious ones such as ( A , A B ) {\displaystyle (A,AB)} ): { ( O , A a ) , ( O , B b ) , ( O , C c ) , ( P , B C ) , ( P , b c ) , ( Q , A C ) , ( Q , a c ) , ( R , A B ) , ( R , a b ) } {\displaystyle \{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\}}

The implication is then ( R , P Q ) {\displaystyle (R,PQ)} —that point R is incident with line PQ.

Famous examples Desargues' theorem holds in a projective plane P if and only if P is the projective plane over some division ring (skewfield) D — P = P 2 D {\displaystyle P=\mathbb {P} _{2}D} . The projective plane is then called desarguesian. A theorem of Amitsur and Bergman states that, in the context of desarguesian projective planes, for every intersection theorem there is a rational identity such that the plane P satisfies the intersection theorem if and only if the division ring D satisfies the rational identity.

Pappus's hexagon theorem holds in a desarguesian projective plane P 2 D {\displaystyle \mathbb {P} _{2}D} if and only if D is a field; it corresponds to the identity ∀ a , b ∈ D , a ⋅ b = b ⋅ a {\displaystyle \forall a,b\in D,\quad a\cdot b=b\cdot a} . Fano's axiom (which states a certain intersection does not happen) holds in P 2 D {\displaystyle \mathbb {P} _{2}D} if and only if D has characteristic ≠ 2 {\displaystyle \neq 2} ; it corresponds to the identity a + a = 0.

References Rowen, Louis Halle, ed. (1980). Polynomial Identities in Ring Theory. Pure and Applied Mathematics. Vol. 84. Academic Press. doi:10.1016/s0079-8169(08)x6032-5. ISBN 9780125998505. Amitsur, S. A. (1966). "Rational Identities and Applications to Algebra and Geometry". Journal of Algebra. 3 (3): 304–359. doi:10.1016/0021-8693(66)90004-4.

Worked examples

Example 1 — a first encounter with Intersection theorem

Start with the simplest possible case. Write down what Intersection theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Intersection theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Intersection theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Intersection theorem

In research: Intersection theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Intersection theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Intersection theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Incidence geometry, Theorems in projective geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Intersection theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Intersection theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Intersection theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Intersection theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Intersection theorem in simple terms?

In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects A and B (for instance, a point and a line). The "theore…

Why does Intersection theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Intersection theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Intersection theorem.