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Visual language (Proportionality and structures (Tesselations (Anomalies…
Visual language
Proportionality and structures
Proportionality between figures: Similarity and Symmetry
Types
Symmetry
Types
Rotational Symmetry
The image is rotated (around a central point) so that it appears 2 or more times.
Reflection Symmetry
One half is the reflection of the other half.
Line of Symmetry
The line that divides the real object with the reflection. It can be in any direction.
A geometric object has symmetry if there is an "operation" or "transformation" that maps the figure/object onto itself.
Similarity
Figures that are the same shape but not necessarily the same size are called similar figures.
How to tell if triangles are similar
Any triangle is defined by six measures (three sides, three angles), but they are similar if:
SSS in same proportion (side side side)
All three pairs of corresponding sides are in the same proportion.
SAS (side angle side)
Two pairs of sides in the same proportion and the included angle equal.
AAA (angle angle angle)
All three pairs of corresponding angles are the same.
Two figures must have the same shape if you want to have a relationship of proportionality between them. If the dimensions are different the relationship between the figures is similar.
Proportionality between figures: Equality
Types
Triangulation
Triangulation is the division of a surface or plane polygon into a set of
triangles, usually with the restriction that each triangle side is entirely shared by two adjacent triangles.
Rotation
Any rotation is a motion of a certain space that preserves at least one point.
Translation
A translation "slides" an object a fixed distance in
a given direction.
Proportion
Proportion means to compare two compositions, to calculate the ratio or the ratio between them.
Equality
Equality is a relationship with a 1: 1 ratio. Two figures are equal when they overlap and coincide on all sides and angles, the figure matches in shape and size. When the shapes are the same length we call it congruents.
Representing Scales
Types
Full Scale
When the dimensions on the drawing are the same than the
actual dimensions of the object.
Reducing Scale
When the dimensions on the drawing are smaller than the
actual dimensions of the object.
Enlarging Scale
When the dimensions on the drawing are bigger than the
actual dimensions of the object.
Scale
The proportion between the drawing and the object.
Scale Factor
The ratio of any two corresponding lengths in two similar geometric figures is called as Scale Factor.
Scale factor can be found in the following scenarios:
Comparing Two Similar Geometric Figures
The scale factor when comparing two similar geometric figures is the ratio of lengths of the corresponding sides.
Size Transformation
In size transformation, the scale factor is the ratio of expressing the amount of magnification.
Scale Drawing
In scale drawing, the scale factor is the ratio of measurement of the drawing compared to the measurement of the original figure.
A scale factor is a number used as a multiplier in scaling.
Tesselations
Semi-regular tesselations
A semi-regular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same. There are only 8 semi-regular tessellations.
Anomalies
Anomaly is a visual resource that consists in a
variation of any element in a tessellation (changing size, shape, direction, position or colour).
The main objective is to attract our attention and
relieve the monotony of repetition.
A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps
Three-dimensional effects
Like any visual expression we can create the sensation of three-dimensional space in tessellations, using various resources: curves, shadows, texture changing...
Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons.
Rep-tiling
A special kind of tiling or tessellation is rep-tiling. Rep-tiles can be joined together to make larger replicas of themselves.
Fundamental domain
The corresponding fundamental domain fact is that for wallpaper tilings they consist of a single tile. Mathematicians call the
smallest piece of a tiling that can be used to rebuild it the fundamental domain.