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#geometry

14 posts12 participants0 posts today
Rasmus

Best effort embedding of a pentagonal {5,5} tiling in 𝑅³ of a surface of infinite genus.

The tiling is self-dual and a partial Cayley surface complex of the group:

⟨ f₁,f₂,f₃,f₄,f₅ ∣ f₁², f₂², f₃², f₄², f₅², f₁f₂f₃f₄f₅, (f₄f₅)², (f₁f₅)², (f₁f₂)³, (f₂f₄f₃f₄)³ ⟩

#math#geometry
Replied in thread
foldworks

@scruss Nice, this version has a few changes to fill the shapes with PENDOWN and PENUP to avoid draw extraneous lines.

TO hx :size
REPEAT 6 [
tri :size FORWARD :size LEFT 60 CLOSE FILL
]
END

TO PT :size
FORWARD :size
RIGHT 150
FORWARD :size
LEFT 30
END

TO superhex :size
REPEAT 6 [
FILLCOLOR "gray" FILLTRANSPARENCY 50 hx :size PU PT :size FORWARD :size LEFT 60 PD
]
END

TO tri :size
REPEAT 3 [
PT :size
]
END

HOME CLEARSCREEN HIDETURTLE
superhex 50

#CreativeCoding#pattern#Sierpinski
Replied in thread
foldworks

@scruss Thanks. Here’s the Sierpinski program adapted to run LibreOffice using the built-in LibreLogo environment help.libreoffice.org/latest/en

One of LibreLogo’s advantages is that the output is vector graphics that can be edited (but rasterised here for convenience).

TO CORNER :S
RIGHT 45 FORWARD :S RIGHT 45
END

TO ONESIDE :S :DIAG :LEVEL
IF :LEVEL = 0 [ STOP ]
ONESIDE :S :DIAG :LEVEL - 1
RIGHT 45 FORWARD :DIAG RIGHT 45
ONESIDE :S :DIAG :LEVEL - 1
LEFT 90 FORWARD :S LEFT 90
ONESIDE :S :DIAG :LEVEL - 1
RIGHT 45 FORWARD :DIAG RIGHT 45
ONESIDE :S :DIAG :LEVEL - 1
END

TO SIERP :S :LEVEL
:DIAG = :S / 1.414
REPEAT 4 [ ONESIDE :S :DIAG :LEVEL CORNER :DIAG ]
END

CS
HT
PU HOME RT 90 BK 100 LT 90 BK 100 PD
SIERP 10 4
FILL

#CreativeCoding#pattern#Sierpinski
Rasmus

Best effort embedding of a pentagonal {5,5} tiling in 𝑅³ of a genus 290 surface.

The tiling is self-dual and a partial Cayley surface complex of the group:

⟨ f₁,f₂,f₃,f₄,f₅ ∣ f₁², f₂², f₃², f₄², f₅², f₂f₅, (f₁f₄)², f₁f₂f₃f₄f₅, (f₂f₃)¹⁷ ⟩

The image and animation show growth step 16 and 17.

(1/2)

#TilingTuesday#geometry#graphtheory
Dani Laura (they/she/he)

(2.5/3) I have been exploring this subject of rectangle partitions from my previous posts.
There is just one rectangle shape which can be divided into two rectangles similar to it, we already now it, it's the rectangle with proportion √2, similar to an A4 paper.
In the case of division of rectangles into three subrectangles similar to it, there are just four rectangle proportions which satisfy it, see figure, they are ordered in decreasing ratio between big and small sides.
First ratio is √3, which is the general case as a rectangle of ratio √n (n in ℕ) can be divided into n similar congruent rectangles.
Second one is √2, this shape can always be divided into n similar rectangles (n in ℕ).
Third ratio is √φ, where φ denotes the golden ratio. This is the only partition which produces three different sizes. I have used this partition in my recent works.
The last one is √(3/2), a partition I have nicknamed "the mouse". I have used this partition in my last artwork "Be safe".

#Mathematics#geometry#TilingTuesday
Rasmus

Slightly perturbed monohedral skew quadragonal tiling embeddable in 𝑅³ as a double cover of an infinite surface.

The tiling is the dual tessellation of a partial Cayley surface complex of the group:

G = ⟨ f₁, f₂, t₁ ∣ f₁², t₁³, f₂², (f₁f₂t₁⁻¹)², (f₁f₂t₁f₂)³, (f₁t₁f₁t₁⁻¹)² ⟩

(1/n)

#math#TilingTuesday#3d
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