The familiar plane geometry of secondary school - figures composed of lines and circles - takes on a new life when viewed as the study of properties that are preserved by special groups of ...

Draw the line of reflection. From a point on the shape, count how far it is vertically from the line of reflection. Count the same amount horizontally on the other side of the line and plot the point.

Both paths led them to the conclusion that certain geometric properties of an object do not change when the object is translated or rotated, expanded or twisted. Geometers discovered that different ...