Kaleidoscopic Tiling by Rachel Duim

Kaleidoscopic Tiling by Rachel Duim
http://filterforge.com/filters/16659.html

WOW!! When downloading and open it is really much better, clever, awesome and amazing how you have been able to create this in such creative way and with huge potential for a great amount of different possible results

here are awesome and amazing animated GIF videos of what can be done

DESCRIPTION OF WHAT DOES MEAN TECHNICALLY ABOUT THE TILINGS USED

Here below is description of each of the tilings used as shown in the file attached above

Below is a detailed explanation of each example shown in the document above, along with additional examples and links to images that illustrate similar tilings.

Background on Uniform and Related Tilings

In tiling theory, a “k‑uniform” tiling is one in which there are exactly k distinct orbits of vertices under the symmetry group. (In a completely vertex‑transitive tiling the tiling is 1‑uniform, and these are sometimes called Archimedean or semiregular tilings.)

Other terms you see include “demiregular” (where only some—but not all—vertices share the same configuration) and “semiregular” (often used interchangeably with Archimedean tilings). The document you provided groups examples by both the number of vertex types (e.g. “3‑uniform”, “4‑uniform”, etc.) and by additional set designations (for instance “Set 1” or “Set 2”) that likely reflect how these tilings were constructed or classified.

Each “Type” in the list is a particular example within a larger catalogue. Below are the explanations:

Type 1: “4‑uniform N‑Uniform Tiles Set 1 Type 8”

What It Means:
This tiling is a member of the 4‑uniform family, meaning that there are four distinct vertex configurations in the tiling.

The label “N‑Uniform” is used here as part of the classification (possibly to distinguish a specific method of construction or normalization) and “Tiles Set 1” groups it with other tilings built by similar rules. “Type 8” indicates that within that group the example is numbered 8.

Additional Example:
An example of a 4‑uniform tiling is one in which four different kinds of vertices occur—for instance, one might see a tiling where vertices have configurations like (3.3.4.3.4), (3.4.3.3.4), etc.

Image Reference:
For an image gallery of k‑uniform tilings (including many 4‑uniform ones), you can visit:
Tilings Gallery – University of Bielefeld

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Type 2: “3‑uniform N‑Uniform Tiles Set 1 Type 1”

What It Means:
Here the tiling belongs to the 3‑uniform family, so there are three distinct types of vertices. It comes fr om the same “Set 1” as Type 1, and “Type 1” indicates it is the first in that subgroup. In a 3‑uniform tiling the different vertex figures might involve different combinations of regular polygons.

Additional Example:
A common 3‑uniform tiling might involve vertices with configurations such as (3.3.4.4) and (3.4.3.4) arranged in an overall periodic pattern.

Image Reference:
For a visual reference, you might check the “k‑uniform tilings” section on Wikipedia:
Wikipedia – Uniform Tilings

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Type 3: “3‑uniform N‑Uniform Tiles Set 1 Type 10”

What It Means:
This is another member of the 3‑uniform tilings in Set 1, but it is numbered “Type 10.” Although it still features three distinct vertex types, its arrangement of polygons is different fr om Type 2.

Additional Example:
There are several known 3‑uniform tilings; for example, a tiling with vertex figures such as (3.3.3.4) alongside another configuration could be found in similar catalogues.

Image Reference:
Again, the Tilings Gallery – University of Bielefeld is a good resource to browse different 3‑uniform tilings.

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Type 4: “7‑uniform N‑Uniform Tiles Set 2 Type 4

[36; 32434; 3342; 3426; 3464; 346; 3636]

What It Means:
This example comes from a more complex family—7‑uniform tilings, meaning there are seven distinct vertex types. It is grouped in “Tiles Set 2” and is the fourth type in that set.

The sequence of numbers in brackets appears to be a shorthand notation that encodes the various vertex configurations or even symmetry‐orders. For instance, “36” might indicate a vertex wh ere a triangle (3) and a hexagon (6) meet, while longer sequences might denote more complex configurations.

Additional Example:
There are other 7‑uniform tilings that can be found in the literature which feature even more varied vertex configurations; these are often used in advanced studies of tiling symmetry.

Image Reference:
For more images and details, one can explore specialized tiling databases such as the one at Tilings Encyclopedia or search for “7-uniform tiling” images online.
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Type 5: “5‑uniform N‑Uniform Tiles Set 2 Type 6”

What It Means:
This tiling is from the 5‑uniform family (featuring five distinct vertex types) and is part of “Tiles Set 2.” It is labeled as Type 6 within that grouping. Its structure is less complex than the 7‑uniform examples but still shows more than the classical 1‑uniform (Archimedean) tilings.

Additional Example:
Similar examples of 5‑uniform tilings include those with mixed vertex figures such as (3.3.4.6) and (3.4.4.6) arranged in a periodic manner.

Image Reference:
Again, resources like the Tilings Gallery – University of Bielefeld provide numerous images of various k‑uniform tilings.
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Type 6: “Regular Wallpaper Group 17, Trihexagonal tiling
[(3.6)2]”

What It Means:
This example is a regular tiling that belongs to one of the wallpaper symmetry groups (group 17 in this classification).
It is known as the trihexagonal tiling because its pattern consists of alternating regular triangles and hexagons.
The notation “[(3.6)2]” indicates that at each vertex the pattern of a triangle and a hexagon repeats twice (i.e. the sequence is triangle–hexagon, triangle–hexagon).

Additional Example:
The trihexagonal tiling is a classic example of a semiregular tiling that appears in many natural and artistic designs.

Image Reference:
A detailed image and explanation can be found here:
Wikipedia – Trihexagonal Tiling

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Type 7: “2‑uniform Demiregular Tiles Type 10”

What It Means:
This tiling is 2‑uniform, meaning it has two different types of vertex configurations.
The term “demiregular” is used to denote that while the tiling has a high degree of symmetry, not every vertex is equivalent (in contrast to fully uniform tilings). “Type 10” is its classification number within this subgroup.

Additional Example:
Other examples of 2‑uniform (or demiregular) tilings might include variations on the snub square tiling or other patterns wh ere two distinct vertex types appear.

Image Reference:
To see more examples, you might search for “2-uniform demiregular tiling” in image galleries such as those on Wolfram MathWorld – Uniform Tilings.

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Type 8: “Semiregular Semi‑regular Tiles Type 3, Truncated Square tiling
[4.8.8]”

What It Means:
This is a semiregular tiling (also known as an Archimedean tiling) in which every vertex has the same configuration. It is identified here as the truncated square tiling.
The vertex configuration [4,8,8] tells us that at each vertex a square (4) and two octagons (8, meet. “Type 3” is the classification number within its subgroup.

Additional Example:
Other well-known semiregular tilings include the truncated hexagonal tiling ([3,12,12]) or the snub hexagonal tiling.

Image Reference:
You can view an image and description here:
Wikipedia – Truncated Square Tiling

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Additional Resources and Image Galleries

For further exploration and visual examples of each type of tiling, consider these resources:

Tilings and Patterns Database:
Tilings Gallery – University of Bielefeld
This site provides a comprehensive collection of k‑uniform tilings and other periodic patterns.

General Information on Uniform Tilings:
Wikipedia – Uniform Tilings
This page offers an overview of the theory and examples, including many of the tilings discussed above.

Mathematical Art and Tessellation:
Tessellation Archive
A resource that shows a variety of artistic and mathematically inspired tilings.

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Summary

Each “Type” corresponds to a specific classification of tilings based on the uniformity of their vertices and the set (or method) by which they were constructed. The numerical labels (like “Type 8” or “Type 10”) identify individual examples within these groups. In addition to the examples in your document, similar tilings can be found in the extensive online galleries and articles referenced above.