Two objects that are the same shape but not the same size are

Two figures are called congruent if they are the same shape and the same size. Two figures are called similar if they are the same shape, but different sizes. More formally, two shapes are similar if they have congruent angles, and corresponding sides are proportional. That means that there is a scale factor number that you multiply each number in the first shape by to get the corresponding side length in the other shape. We often solve for missing side lengths using proportions.

Question

Asked 17th May, 2016

James F Peters

• University of Manitoba

What do we mean by shape and when do two objects have similar shapes?

By definition, a shape is an external form or appearance of something. This is how the Oxford English Dictionary defines the term shape. But then this definition raises more questions than it answers.

In his Essentials of Topology with Applications, CRC Press, 2010, Steven G. Krantz asks whether a ruler and a sheet of paper have the same shape, since both are rectangles.

We might also ask the following related questions.

Does a donut have the same shape as a wedding ring, since each one has a hole in its center?

For that matter, do all objects with a single hole in their centers have the same shape?

Is the concept of hole part of the concept of shape? In other words, do we need to take into account the presence or absence of holes in every shape?

There are many different types of shapes in Physical Science. For example, a Wulff shape is anan equilibrium minimal surface for a crystal or drop which has the least anisotropic surface free energy for a given volume. Wulff shape are explained in

http://mathworld.wolfram.com/WulffShape.html

The theory of shape is a central topic in Mathematics. For example, Karol Borsuk introduced the theory of shape in his 1970 lectures:

http://journals.cambridge.org/download.php?file=%2FBAZ%2FBAZ22_02%2FS000497270000647Xa.pdf&code=4512e52047bf2b8367d2aaa56a4c8e16

http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0481.0490.ocr.pdf

For Borsuk, shape theory is the focus of geometric topology, which is a study of the topological properties of metrizable spaces.

http://mathworld.wolfram.com/MetrizableTopology.html

Shape theory is also closely related to what are known as retracts.

http://mathworld.wolfram.com/Retract.html

Long before the study of shapes entered into the picture in the Physical Sciences and in Mathematics, shapes were the focus of the Fine Arts (painting and sculpture) and Philosophy. Capturing shapes is a central activity in painting. A classical example is the chiaroscuro effectusing various forms of highlighting objects:

http://painting.about.com/od/oldmastertechniques/a/sfmuato_chiaros.htm

And shapes were (and still are) of great interest in Philosophy, The classical example an interest in external forms can be found in the works of Plato and Artistotle.

What we mean by shape? is an open question. A related open question concerns similar shapes. When do objects have similar shapes?

Shape

Topology

Geometry and Topology

Physical Sciences