Definition:Geometric Figure
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Definition
A geometric figure is intuitively defined as a set of points and lines in space.
As Euclid defined it:
- A figure is that which is contained by any boundary or boundaries.
(The Elements: Book I: Definition $14$)
The boundary may or may not be included in a particular figure. If this is important (and in the study of topology it usually is), then whether it is included or not needs to be specified.
Plane Figure
A plane figure is a geometric figure embedded in the plane.
Three-Dimensional Figure
A three-dimensional figure is a geometric figure which can not be embedded in the plane, but which can be embedded in three-dimensional space.
A common term for this is solid figure but this is a misnomer as it is usual to be studying the surfaces of such figures, which themselves are not solid as such.
Rectilineal Figure
As Euclid defined it:
- Rectilineal figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multi-lateral those contained by more than four straight lines.
(The Elements: Book I: Definition $19$)
The usual name for a plane rectilineal figure is polygon.
The usual name for a 3-dimensional rectilineal figure is polyhedron.
Diameter
The diameter of a geometric figure is the greatest length that can be formed between two opposite parallel straight lines that can be drawn tangent to its boundary.
Compare with the diameter of a circle, whose definition is consistent with this.
Note
The diameter of a parallelogram is not consistent with this definition.
