There are three polyhedra that tile 3D space:

All physical matter is composed of geometry within geometry – a fractal-holographic matrix of interconnected consciousness and information. This is key! At the molecular structure of all matter exists beautifully ordered geometric lattices upon which life is built. These tilings also appear in crystal, molecular and cellular structures in nature. These tilings are the basis for Celtic & Islamic art. They allow for more than 1 polygon and for 2 vertex situations.

Schlafli symbols are a more compact way to describe tilings.įor instance, an equilateral triangle is cannot be tiled in 2D or 3D. When naming, always start with the polygon with the least number of sides. To name a tessellation, go around a vertex (corner point) and write down how many sides each polygon has, in order.įor example, a square tiling would be 4.4.4.4.Ī regular hexagonal tiling would be 6.6.6.” 3Ī semiregular tiling of hexagons and triangles (snub hexagonal tiling) would be 3.3.3.3.6. “When the tessellation is made of regular polygons, the most common notation is the vertex configuration, which is simply a list of the number of sides of the polygons around a vertex. The part that repeats is called a ’tile’ regardless of its shape. We have seen these with Penrose pentagonal tilings in Article 54. There are also semiregular tilings with regular tiles of more than one shape and with every corner identically arranged.Ī tiling that lacks a repeating pattern is called ‘non-periodic’. There are regular tilings with regular polygonal tiles of all the same shape. Tessellation occurs when the same regular polygon is used to fill the plane, leaving no gaps, nor overlapping. Some of these manipulations can be seen in dual uniform tilings, such as displacement, reflection and roration. Various combinations of the above can be used. Stretching – making the tile longer or wider without tearing or breaking Scaling – keeping the same shape but changing the relative size of a tile Rotation – the action of rotating a tile around an axis or center Reflection – inverting a tile with respect to a plane, each element being transferred perpendicularly through the plane to a point the same distance the other side of it. Symmetrical manipulations are as follows:ĭisplacement (translation) – movement from place or position a certain distance in a certain direction – leaves it appearing ‘the same’. In addition, symmetry principles are characterized by a quietude, a stillness that is somehow beyond the bustling world yet, in one way or another, they are almost always involved with transformation, disturbance, or movement.”2 But in itself symmetry is unlimited there is nowhere that its principles do not penetrate. Symmetry precepts are always involved with categorization, with classification and observed regularities in short, with limits. It creates three-corner 120-degree joints, which provide the triangle’s strength and structural excellence to the package.”1ĭavid Wade tells us, “Any notion of symmetry is completely entangled with that of asymmetry we can scarcely conceive of the former without invoking thoughts of the latter.

“Hexagonal tessellation extends the circle’s principle of equality in all directions in a balanced sharing of space, materials, time, and energy. When the circles are converted to hexagons, no gaps are left at all. When circles fill space in a hexagonal pattern, they leave curved triangular gaps between them. Larger curved diamond shaped gaps remain. Curved triangular gaps remain.Ĭircular packing in a square pattern. We will start with one important quality of the hexagon – that of hexagonal packing.Ĭircles, spheres & cylinders fill space more efficiently in a hexagonal pattern than a square pattern.Ĭircular packing in a hexagonal pattern. We will expand from just the hexagon to cover 2D and 3D tessellations and geometric tiling. We will now discuss one important aspect of the hexagon – that of tessellations.

Octagon tessellation