mathematics

Penrose triangle

Penrose triangle 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 Penrose triangle rather than just read about it. In short: The Penrose triangle, also known as the Penrose tribar, the impossible tribar, or the impossible triangle, is a triangular impossible object, an optical illusion consisting of an object that can be depicted in a perspective drawing. It cannot exist as a solid object in ordinary three-dimensional Euclidean space, although its surface can be embedded isometrically (bent but not stretched) in five-dimensional Euclidean…

Key takeaways

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

Reference excerpt

The Penrose triangle, also known as the Penrose tribar, the impossible tribar, or the impossible triangle, is a triangular impossible object, an optical illusion consisting of an object that can be depicted in a perspective drawing. It cannot exist as a solid object in ordinary three-dimensional Euclidean space, although its surface can be embedded isometrically (bent but not stretched) in five-dimensional Euclidean space. It was first created by the Swedish artist Oscar Reutersvärd in 1934. Independently from Reutersvärd, the triangle was devised and popularized in the 1950s by psychiatrist Lionel Penrose and his son, the mathematician and Nobel Prize laureate Roger Penrose, who described it as "impossibility in its purest form". It is featured in the lithograph Waterfall (1961) by artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it.

Description

The tribar/triangle appears to be a solid object, made of three straight beams of square cross-section that meet pairwise at right angles at the vertices of the triangle they form. The beams may be broken, forming cubes or cuboids. This combination of properties cannot be realized by any three-dimensional object in ordinary Euclidean space. Such an object can exist in certain Euclidean 3-manifolds. A surface with the same geodesic distances as the depicted surface of the tribar, but without its flat shape and right angles, are to be preserved, can also exist in 5-dimensional Euclidean space, which is the lowest-dimensional Euclidean space within which this surface can be isometrically embedded. There also exist three-dimensional solid shapes each of which, when viewed from a certain angle, appears the same as the 2-dimensional depiction of the Penrose triangle, such as the sculpture "Impossible Triangle" in East Perth, Western Australia The term "Penrose Triangle" can refer to the 2-dimensional depiction or the impossible object. If a line is traced around the Penrose triangle, a 4-loop Möbius strip is formed.

Creation of the Penrose triangle from partial figures

If the left part of the figure is moved parallel to the right until its upper horizontal edge coincides with the upper horizontal edge of the middle part of the figure, the Penrose triangle (right) is created by overlapping the two parts. The first two partial views of the Penrose triangle are individually perceptible, whereas the resulting tribar represents an impossible figure.

Depictions

M. C. Escher's lithograph Waterfall (1961) depicts a watercourse that flows in a zigzag along the long sides of two elongated Penrose triangles, so that it ends up two stories higher than it began. The resulting waterfall, forming the short sides of both triangles, drives a water wheel. Escher points out that in order to keep the wheel turning, some water must occasionally be added to compensate for evaporation. A third Penrose triangle lies between the other two, formed by two segments of waterway and a support tower.

Sculptures

See also Impossible cube Impossible trident Shepard elephant Penrose stairs

References

External links An article about impossible triangle sculpture in Perth Escher for Real constructions

Illustrations

Worked examples

Example 1 — a first encounter with Penrose triangle

Start with the simplest possible case. Write down what Penrose triangle 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 Penrose triangle 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 Penrose triangle 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 Penrose triangle

In research: Penrose triangle 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 Penrose triangle 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: Penrose triangle is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1934 introductions, Impossible objects, Roger Penrose, so understanding it makes those chapters shorter.
In everyday life: Look for Penrose triangle 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Penrose triangle in 20 minutes

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

Frequently asked questions

What is Penrose triangle in simple terms?

Why does Penrose triangle 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 Penrose triangle?

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 Penrose triangle.