From Wikipedia, the free encyclopedia
Impossible object
Penrose triangle
The
Penrose triangle
, also known as the
Penrose tribar
, the
impossible tribar
,
[1]
or the
impossible triangle
,
[2]
is a triangular
impossible object
, an
optical illusion
consisting of an object which can be depicted in a perspective drawing. It cannot exist as a solid object in ordinary three-dimensional Euclidean space, although it can be embedded isometrically in five-dimensional Euclidean space.
[3]
It was first created by the Swedish artist
Oscar Reutersvard
in 1934.
[4]
Independently from Reutersvard, 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".
[5]
It is featured prominently in the works of artist
M. C. Escher
, whose earlier depictions of impossible objects partly inspired it.
Description
[
edit
]
A rotating Penrose triangle model to show illusion. At the moment of illusion, there appears to be a pair of purple faces (one partially occluded) joined at right angles, but these are actually parallel faces, and the partially occluded face is internal, not external.
The tribar/triangle appears to be a
solid
object, made of three straight beams of square cross-section which 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
,
[6]
and can also exist in 5-dimensional Euclidean space,
[7]
which is the lowest-dimensional Euclidean space within which the Penrose triangle can be 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 on this page (such as ? for example ? the adjacent image depicting a sculpture in
Perth
,
Australia
). The term "Penrose Triangle" can refer to the 2-dimensional depiction or the impossible object itself.
If a line is traced around the Penrose triangle, a 4-loop
Mobius strip
is formed.
[8]
Depictions
[
edit
]
A 3D-printed version of the Reutersvard Triangle illusion
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
.
Sculptures
[
edit
]
See also
[
edit
]
References
[
edit
]
- ^
Pappas, Theoni
(1989). "The Impossible Tribar".
The Joy of Mathematics: Discovering Mathematics All Around You
. San Carlos, California: Wide World Publ./Tetra. p. 13.
- ^
Brouwer, James R.; Rubin, David C. (June 1979). "A simple design for an impossible triangle".
Perception
.
8
(3): 349?350.
doi
:
10.1068/p080349
.
PMID
534162
.
S2CID
41895719
.
- ^
https://www.maplesoft.com/mapleconference/resources/54_Zeng_IsometricEmbedding_slides.pdf
- ^
Ernst, Bruno (1986). "Escher's impossible figure prints in a new context". In
Coxeter, H. S. M.
; Emmer, M.;
Penrose, R.
; Teuber, M. L. (eds.).
M. C. Escher Art and Science: Proceedings of the International Congress on M. C. Escher, Rome, Italy, 26?28 March, 1985
. North-Holland. pp. 125?134.
See in particular p. 131.
- ^
Penrose, L. S.
;
Penrose, R.
(February 1958). "Impossible objects: a special type of visual illusion".
British Journal of Psychology
.
49
(1): 31?33.
doi
:
10.1111/j.2044-8295.1958.tb00634.x
.
PMID
13536303
.
- ^
Francis, George K. (1988). "Chapter 4: The impossible tribar".
A Topological Picturebook
. Springer. pp. 65?76.
doi
:
10.1007/978-0-387-68120-7_4
.
ISBN
0-387-96426-6
.
See in particular p. 68, where Francis attributes this observation to
John Stillwell
.
- ^
https://www.maplesoft.com/mapleconference/resources/54_Zeng_IsometricEmbedding_slides.pdf
- ^
Gardner, Martin (August 1978). "Mathematical Games: A Mobius band has a finite thickness, and so it is actually a twisted prism".
Scientific American
.
239
(2): 18?26.
doi
:
10.1038/scientificamerican1278-18
.
JSTOR
24960346
.
- ^
Федоров, Ю. (1972).
"Невозможное-Возможно"
.
Техника Молодежи
.
4
: 20?21.
External links
[
edit
]
|
---|
Illusions
| | |
---|
Popular culture
| |
---|
Related
| |
---|
|
---|
Books
| |
---|
Coauthored books
| |
---|
Academic works
|
- Techniques of Differential Topology in Relativity
(1972)
- Spinors and Space-Time: Volume 1, Two-Spinor Calculus and Relativistic Fields
(with
Wolfgang Rindler
) (1987)
- Spinors and Space-Time: Volume 2, Spinor and Twistor Methods in Space-Time Geometry
(with Wolfgang Rindler) (1988)
|
---|
Concepts
| |
---|
Related
| |
---|