The mathematical concepts of wizzle are Isometries and Tessellations.

Isometries are geometric transformations that do not alter  the shape (distance-preserving), meaning various types of movements of a shape in the plane. When we talk about geometric transformations, we often use the more vague and general term symmetries. The basic types of isometries include translation, reflection, glide-reflection, and rotation: 

Tessellations, or tilings, are arrangements of shapes (same or different  ones ) that completely cover the plane (with a periodic or a-periodic way) without any gaps or overlaps. 

The tiles of wizzle are designed in such a way as to have various types of symmetries and to be able to be placed next to each other, creating a tessellation (tiling) on a plane. This tessellation is theoretically infinite and can practically expand as much as desired, while maintaining the same symmetries as the wizzle tiles.

For more mathematical details about the symmetries of each wizzle tile, you can find information in the product's specifications.

Wizzle is about symmetries! Symmetry is the core and the origin of geometric thinking. Geometry, and scientific thinking in general, operates based on the search for regularities in the phenomena it seeks to interpret, and symmetries are the ones that create regularities in the natural world and in the world of scientific ideas created by man.

The  wizzle tiles are designed based on mathematical rules, which are transferred to the tilings created with these tiles. When someone solves wizzle, they are practically implementing these mathematical rules, even if they are not aware of it or do not have conscious knowledge of these rules. By exploring the different patterns that can be created with wizzle and through repetition, the user becomes familiar with specific properties of geometric shapes and develops a visual literacy that forms the background of geometric thinking and the ability to see in a geometric shape what the eye of an experienced mathematician can perceive. The power of wizzle lies in visualizing the mathematical rules it encompasses and enabling the user to negotiate these mathematical rules without the need for mathematical symbols and formal mathematical language.

From the Open Educational Practice "Symmetries, From Art to Geometry", Seal of Quality for Best Educational Practice, 1st Prize in the Open Educational Practices Competition.

Symmetries are of fundamental importance to humans. Humans are built with symmetries, surrounded by symmetries, and operate and think based on symmetries. Wizzle is symmetries!

Humans are built with symmetries:

The human body has bilateral symmetry (reflection). Most living organisms that live on land, in the sea, and in the air have the same bilateral symmetry. Organisms that developed this type of symmetry had the ability to move in search of food and, therefore, had a greater chance of survival, and that is why they prevailed. Bilateral symmetry aids in balance and movement. It is worth noting that in all living organisms that have bilateral symmetry, the level of symmetry is always vertical, following the direction of gravity and perpendicular to the direction of movement.

Humans are surrounded by symmetries:

Symmetries are patterns of development in the natural world. All leaves have bilateral symmetry (reflection), while flowers often develop with radial symmetry (rotations). The symmetries that flowers develop create regularity within the chaos of the natural world, attracting pollinators such as bees, butterflies, and other insects that visit the flowers and carry their pollen, thereby increasing their chances of survival and reproduction.

The symmetries possessed by animal and plant organisms help conserve the information needed for their development, stored in their DNA. If a daisy has 30 identical petals, it is sufficient to have the information for the development of one petal stored in the DNA, which will be repeated for each petal.

Symmetries play an important role in the microcosm. The symmetries with which atoms and molecules are organized determine the properties of materials. For example, precious diamonds and inexpensive graphite have exactly the same chemical composition—they are both made of carbon. The symmetries of their crystalline structures are responsible for their completely different properties.

Humans   function with symmetries:

There are many dipoles in space (east-west, north-south, front-back, right-left, up-down) that are related to spatial skills used by humans to orient and move.

Man imitates the symmetries observed in his environment in his artifacts and technology. The hand axe, the primary tool of prehistoric man, had remarkably precise bilateral symmetries. All means of transportation on land, sea, and air designed and constructed by humans possess the same bilateral symmetry that humans themselves possess, which serves balance and movement. The wheel, propellers, and helix imitate the radial symmetry found in flowers. Many everyday objects (forks, spoons, knives, pliers, tongs, tweezers, grinders, combs, fans, etc.) possess symmetries that serve their functionality.

Humans think with symmetries:

Dr. Stanislas Dehaene, cognitive neuroscientist and professor at the Collège de France, argues that what differentiates the human mind and makes it unique are the innate geometric intuitions it possesses.

Researchers at the University of OLVI argue that thinking processes do not occur solely in the brain. There are thinking processes that take place in the fingertips and are related to the geometric properties of objects perceived by touch.

All civilizations that flourished on the planet, even those that had no interactions with each other, developed the same symmetries in decorative arts (translation, reflection, glide reflection, and rotation). This means that symmetries constitute a universal language.

Archaeological findings in Mezin, Ukraine (20,000 BCE), have engraved decorations with linear non-representational designs (meanders) that possess the symmetry of double rotation. It seems that symmetries are the first recorded (written) mathematical ideas expressed in the language of art, 17,000 years before the first written language. The fact that prehistoric humans designed not an animal possessing symmetry, but the idea of symmetry itself, represents an example of abstract thinking. Perhaps it is a cognitive leap of prehistoric humans, perhaps prehistoric humans had the ability to think with symmetries. In any case, it constitutes a step towards scientific thinking.