mathematics

Lotschnittaxiom

Lotschnittaxiom is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lotschnittaxiom rather than just read about it. In short: The Lotschnittaxiom (German for "axiom of the intersecting perpendiculars") is an axiom in the foundations of geometry, introduced and studied by Friedrich Bachmann. It states: Perpendiculars raised on each side of a right angle intersect.

Key takeaways

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

Reference excerpt

The Lotschnittaxiom (German for "axiom of the intersecting perpendiculars") is an axiom in the foundations of geometry, introduced and studied by Friedrich Bachmann. It states:

Perpendiculars raised on each side of a right angle intersect. Bachmann showed that, in the absence of the Archimedean axiom, it is strictly weaker than the rectangle axiom, which states that there is a rectangle, which in turn is strictly weaker than the Parallel Postulate, as shown by Max Dehn. In the presence of the Archimedean axiom, the Lotschnittaxiom is equivalent with the Parallel Postulate.

Equivalent formulations As shown by Bachmann, the Lotschnittaxiom is equivalent to the statement Through any point inside a right angle there passes a line that intersects both sides of the angle. It was shown in that it is also equivalent to the statement The altitude in an isosceles triangle with base angles of 45° is less than the base. and in that it is equivalent to the following axiom proposed by Lagrange: If the lines a and b are two intersecting lines that are parallel to a line g, then the reflection of a in b is also parallel to g. As shown in, the Lotschnittaxiom is also equivalent to the following statements, the first one due to A. Lippman, the second one due to Henri Lebesgue

Given any circle, there exists a triangle containing that circle in its interior. Given any convex quadrilateral, there exists a triangle containing that convex quadrilateral in its interior. Three more equivalent formulations, all purely incidence-geometric, were proved in: Given three parallel lines, there is a line that intersects all three of them. There exist lines a and b, such that any line intersects a or b. If the lines a_1, a_2, and a_3 are pairwise parallel, then there is a permutation (i,j,k) of (1,2,3) such that any line g which intersects a_i and a_j also intersects a_k.

In Bachmann's geometry of line-reflections Its role in Friedrich Bachmann's absolute geometry based on line-reflections, in the absence of order or free mobility (the theory of metric planes) was studied in and in.

Connection with the Parallel Postulate As shown in, the conjunction of the Lotschnittaxiom and of Aristotle's axiom is equivalent to the Parallel Postulate.

References

Sources

Worked examples

Example 1 — a first encounter with Lotschnittaxiom

Start with the simplest possible case. Write down what Lotschnittaxiom 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 Lotschnittaxiom 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 Lotschnittaxiom 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 Lotschnittaxiom

In research: Lotschnittaxiom 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 Lotschnittaxiom 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: Lotschnittaxiom is common in secondary-school and first-year university syllabi. It links to neighbouring topics Foundations of geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Lotschnittaxiom 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.

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How to study Lotschnittaxiom in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lotschnittaxiom 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 Lotschnittaxiom out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lotschnittaxiom in simple terms?

The Lotschnittaxiom (German for "axiom of the intersecting perpendiculars") is an axiom in the foundations of geometry, introduced and studied by Friedrich Bachmann. It states: Perpendiculars raised on each side of a right angle intersect.

Why does Lotschnittaxiom 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 Lotschnittaxiom?

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 Lotschnittaxiom.