Nummolt Blog - Mathematical engine mill

Tuesday, December 10, 2024

Analysis of 6 tessellations with lizards in Escher's drawings.

6 Escher's tessellations with lizards:

1 - Lizard tessellation classified here as Heesch 04 CCCC:

Searching the four nodes and four articulations of the 4 Heesch's 'C' edges:

(Circles in all nodes)
(Stars in center of 180º rotation of C edges)

Rebuilding this lizard tile with Explore Tessellations:

Tessellation based on CCCC:

2 - Lizard tessellation classified here as Heesch 09 C3C3C3C3C3C3:

Searching the six nodes in the original drawing:

(Double circle in the nodes next to edges under rotation of 120º)

Rebuilding lizard C3C3C3C3C3C3 tile with Explore Tessellations:

Tessellation based on C3C3C3C3C3C3:

3 - Lizard tessellation classified here as Heesch 14 CC4C4:

Searching the 3 nodes in the original drawing:

(Double circle in the vertex of 90º rotation of 2 C4 edges)
(Cross inside circle in center of 180º rotation of C edge)

Rebuilding lizard CC4C4 tile with Explore Tessellations:

Tessellation based on CC4C4:

4 - Lizard tessellation classified here as Heesch 16 CC4C4C4C4:

Searching the 5 nodes in the original drawing:

(Double circle in the vertex of 90º rotation of 2 C4 edges)
(Cross inside circle in center of 180º rotation of C edge)

Rebuilding lizard CC4C4C4C4 tile with Explore Tessellations:

Tessellation based on CC4C4C4C4:

5 - Lizard tessellation classified here as Heesch 22 CCGG:

Searching the 4 nodes in the original drawing:

(Circles in all nodes)
(Double circle in articulation of two G flipped edges)
(Stars in center of 180º rotation of C edges)

Rebuilding lizard CCGG tile with Explore Tessellations:

Tessellation based on CCGG:

6 - Lizard tessellation classified here as Heesch 22 CGCG:

Searching the 4 nodes in the original drawing:

(Circles in all nodes)
(Stars in center of 180º rotation of C edges)

Rebuilding lizard CCGG tile with Explore Tessellations:

Tessellation based on CGCG:

- - - - - - - - - - - - - - - - - - - - -

(The classification of Escher Tiles has been done here under the Heesch Classification that you can find at: http://www.eschertile.com/ )

(The rebuild of Escher's tiles has been done here with the android app "Explore Tessellations": https://play.google.com/store/apps/details?id=com.nummolt.heesch.escher.tiles.tessellations )

YOUTUBE VIDEO: 6 LIZARDS.

https://www.youtube.com/watch?v=JtH6ZOWBAM0

Sunday, December 1, 2024

Exploring Escher's "Two Birds" (1938) with Explore Tessellations.

Escher's "Two birds" (1938)

At first glance at the drawing we might think that the tile from which Escher starts is one of the birds, repeating itself through translations, symmetries and rotations.

We can start by looking for the nodes where several tiles join together:

And we could think of a Heesch 22 type tile with two sides of type G formed by the same type of line but symmetrical, and two sides of type C with their respective 180º turns.

Like this:

But this is not the solution, because 180º symmetries put the birds upside down. And in Escher tessellation there are no upside down birds.

* * * *

We need to look at Escher's drawing again. It seems that the two birds are neither equal nor symmetrical to each other:

This is only possible if the two different birds are part of the same tile:

Now this tile is a hexagon containing two birds with three translations to tessellate. The Heesch TTTTTT type. (Heesch 2)

Now the result is similar to the original Escher drawing:

In 1938 Escher was experimenting with tiles with two figures inside.

The same birds appear developed and gradually correcting their differences in Escher's famous drawing: "Day and Night"

Day and night. M.C. Escher 1938

* * *

(Graphics made with "Explore Tessellations" Android app:
https://play.google.com/store/apps/details?id=com.nummolt.heesch.escher.tiles.tessellations)

Tuesday, June 14, 2022

Adding Unit Fractions - Google Play

Adding Unit Fractions

https://play.google.com/store/apps/details?id=com.nummolt.fractions.add

The app proposes 21 challenges to overcome.
Obtaining the proper fractions listed at the top of the application, adding two or three unit fractions.

Each proposed proper fraction has a variable number of solutions.
And different levels of difficulty

You can not repeat unit fractions with the same value.
In the app you'll find a button to delete all the solutions found in the current problem, and to start from scratch.
The lower unit fraction used in this app is 1/28.

The app allow to decompose 2/3, 3/4, 3/5, 4/5, 5/6, 3/7, 4/7, 5/7, 3/8, 5/8, 7/8, 4/9, 5/9, 7/9, 8/9, 3/10, 7/10, 9/10.

The app works right in Android TV devices.

3/7 Only 1 solution with 1/28 as the tiniest available unit fraction