mathematics

Geometry and the Imagination

Geometry and the Imagination 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 Geometry and the Imagination rather than just read about it. In short: Geometry and the Imagination is the English translation of the 1932 book Anschauliche Geometrie by David Hilbert and Stefan Cohn-Vossen. The book was based on a series of lectures Hilbert made in the winter of 1920–21.

Key takeaways

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

Reference excerpt

Geometry and the Imagination is the English translation of the 1932 book Anschauliche Geometrie by David Hilbert and Stefan Cohn-Vossen. The book was based on a series of lectures Hilbert made in the winter of 1920–21. The book is an attempt to present some then-current mathematical thought to "contribute to a more just appreciation of mathematics by a wider range of people than just the specialists." It differentiates between two tendencies in mathematics and any other scientific research: on the one hand, toward abstraction and logical relations, correlating the subject matter in a systematic and orderly manner, and on the other hand an intuitive approach, which moves toward a more immediate grasp of and a "live rapport" with the same material. Further he asserts that intuitive understanding actually plays a major role for the researcher as well as anyone who wishes to study and appreciate Geometry.

Contents Topics covered by the chapters in the book include the Leibniz formula for π, configurations of points and lines with equally many points on each line and equally many lines through each point, curvature and non-Euclidean geometry, mechanical linkages, the classification of manifolds by their Euler characteristic, and the four color theorem.

Response The Mathematical Association of America said about the book, "this book is a masterpiece — a delightful classic that should never go out of print". Physics Today called it "a readable exposition of modern geometry and its relation to other branches of mathematics". The Scientific Monthly said about it "has been a classic for twenty years . . . Although it deals with elementary topics, it reaches the fringe of our knowledge in many directions".

References

External links Geometry and the Imagination at the Internet Archive

Worked examples

Example 1 — a first encounter with Geometry and the Imagination

Start with the simplest possible case. Write down what Geometry and the Imagination 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 Geometry and the Imagination 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 Geometry and the Imagination 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 Geometry and the Imagination

In research: Geometry and the Imagination 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 Geometry and the Imagination 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: Geometry and the Imagination is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1932 non-fiction books, 1952 non-fiction books, Books about mathematics, so understanding it makes those chapters shorter.
In everyday life: Look for Geometry and the Imagination 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 Geometry and the Imagination in 20 minutes

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

Frequently asked questions

What is Geometry and the Imagination in simple terms?

Why does Geometry and the Imagination 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 Geometry and the Imagination?

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 Geometry and the Imagination.