Stephanie Erickson and Gary Felder

CurvedLand is an applet for showing what the world would look like with
different geometry. It is named CurvedLand in tribute to the science
fiction novel,
*Flatland*, by Edwin Abbott, which describes the adventures of a
two-dimensional being who is visited by a stranger from the third dimension.

One of the central ideas of Einstein's theory of relativity is that space and time curve in response to the matter and energy within them. A curved space is one that doesn't obey the usual laws of Euclidean geometry: the angles of a triangle don't generally add up to 180 degrees, the circumference of a circle isn't pi times the diameter, parallel lines can either converge towards each other or move apart, and so on.

Since the geometry we observe is very close to Euclidean, however, it is hard for most of us to picture what this difference would mean physically. If you draw a circle and a diameter, how could the ratio be anything other than pi? To answer this question, imagine that as you move around in space the shapes of objects appear to distort. This is what happens in curved space. If you draw a circle around yourself and then start walking around it to pace out the circumference, it will look to you like you are walking along a constantly changing ellipse.

CurvedLand illustrates this distortion as it would appear in a two-dimensional curved space. The structure is similar to a mapping program. You can place objects of different shapes in different places in the world and then move around the space to see what they look like from different perspectives.

Click here for instructions on the use of the CurvedLand applet. Click here for more background on the physics and math of curved geometry in general. Click here for more information on the particular model being illustrated in CurvedLand, including a discussion of what it means to use a "bird's eye view" of a two dimensional space.

If you have further questions, comments, suggestions, or requests please feel free to email Gary Felder.

- Tutorial 0: Finding your way around CurvedLand
- Tutorial 1: Lines in Curved Space
- Tutorial 2: Triangles in Curved Space
- Tutorial 3: Circles in Curved Space

Copyright (C) 2010 Stephanie Erickson, Gary Felder

This material is based upon work supported by the National Science Foundation under Grant No. #0757746. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

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