Coxeter on Hyperbolic Geometry & Escher's Circle Limit

Tessellations (teselados) are geometric patterns that cover a plane without leaving spaces between pieces. They are commonly found in floors, walls, staircases, houses, bathrooms, kitchens, buildings, museums, and other constructions. Tessellations can also be found in nature, such as in honeycombs, and in objects like doors, windows, gates, fences, and other items. They can serve decorative or artistic purposes. Notable figures who studied tessellations include physicist and mathematician Roger Penrose (who won the 2020 Nobel Prize in Physics) and Dutch artist M.C. Escher.

Tessellations are 2D geometric patterns made by repeating shapes that fit perfectly together without gaps or overlaps; effective tessellation drawing involves analyzing the reference pattern to identify underlying line structures (such as horizontal, vertical, and diagonal lines at specific angles like 60° or 45°), drawing these lines systematically rather than individual shapes, and using light initial strokes that can be easily corrected.

A tessellation is an arrangement of geometric tiles that completely covers a plane with no gaps or overlaps, and it can be created using two methods: (1) starting with a square and cutting/taping equal amounts from opposite sides to create complex shapes, or (2) starting with an equilateral triangle and applying similar cut-and-tape rules to opposite sides and halves; remarkably, any arbitrary quadrilateral can also tessellate by making copies, flipping them, and arranging them in alternating rows.

This lesson covers tessellations—the art of filling a flat surface with shapes without gaps or overlaps. MC Escher's famous works demonstrate artistic applications, while real-world examples include brick walls, tiled floors, and beehives. Regular tessellations use only one regular polygon (triangles, squares, hexagons), determined by checking if interior angles divide evenly into 360°. Three main transformation types exist: translation (sliding), rotation (turning), and glide reflection (sliding then flipping). Universally, all triangles and quadrilaterals tessellate due to their angle sums (180° and 360° respectively). Even regular polygons that cannot tessellate alone (like octagons) can combine with other shapes to create valid tessellations.

A tessellation is a mosaic with regular or semi-regular repetition of fragments that leave no spaces and do not overlap. To create one, start with a 5cm square base. Draw two shapes on opposite sides, cut them out, and attach them to the free sides using tape. The shapes must fit together like a puzzle, requiring slight rotation for proper alignment. Once the base unit is complete, repeat it across the entire surface by first drawing all units in pencil, then adding the final design in ink.

This comprehensive section covers the three main types of geometric transformations: translation, rotation, and reflection. Translation involves moving points or shapes along a specific direction and distance, with rules for points (x+a, y) for right movement, (x-a, y) for left, (x, y+a) for up, and (x, y-a) for down. For lines Ax + By + C = 0, translation in positive direction requires subtracting from the coordinate. Rotation involves turning points around a center by a specific angle, with 90° counterclockwise rotation around origin following quadrant-based rules: first quadrant (a,b) becomes (-b,a), second quadrant (a,b) becomes (b,-a), third quadrant (a,b) becomes (-b,-a), fourth quadrant (a,b) becomes (b,a). Reflection creates mirror images across lines or points, with point-to-point reflection using (2x2-x1, 2y2-y1) and point-to-axis reflection following specific coordinate changes. Special cases include lines with slope ±1, where reflection simplifies to replacing coordinates. The section emphasizes that all three transformation types can appear in exam questions, requiring careful attention to direction, center, and axis.

A transformation is a function that maps elements from a starting set to an arriving set. In geometry, a transformation associates each point of a plane with a unique image point. The three fundamental geometric transformations are: (1) Central symmetry - reflection through a center point, (2) Axial symmetry - reflection across a line, and (3) Translation - sliding along a vector. Each transformation has characteristic elements: central symmetry has a center point, axial symmetry has an axis of symmetry, and translation has a translation vector.

This comprehensive section covers the three fundamental geometric transformations: Reflection (انعكاس) involves flipping an object across a line called the axis of reflection, where the image and original have equal distances from the axis. Translation (انتقال) moves a shape by a specific distance in a particular direction without changing its orientation. Rotation (دوران) turns a shape around a fixed center point by a certain angle. The instructor then covers the coordinate system with x-axis and y-axis, explaining that each point has coordinates (x, y). The key rules are: reflecting across the x-axis keeps x unchanged while changing y's sign, and reflecting across the y-axis keeps y unchanged while changing x's sign. Points on the axes remain unchanged when reflected across those same axes. The section demonstrates practical application by reflecting triangles and parallelograms with given coordinates across coordinate axes. The key properties of reflection are covered: it preserves lengths of line segments (congruence), parallelism between lines, angle measures, and collinearity (points on lines remain on reflected lines).

This section covers standard geometric transformations that change only the position of a function's graph without altering its shape. Vertical translation (انسحاب رأسي) moves the graph up or down when applied outside the function: positive values move up, negative values move down. Horizontal translation (انسحاب أفقي) moves the graph left or right when applied inside the function: negative values move right, positive values move left. Reflection across the x-axis (انعكاس حول محور اكس) is achieved by placing a negative sign outside the function. Reflection across the y-axis (انعكاس حول محور واي) is achieved by placing a negative sign inside the function. These transformations are fundamental for understanding how functions can be shifted and flipped.

This section introduces the fundamental concepts of geometric transformations in plane geometry. A transformation (phép biến hình) is defined as a rule that maps each point M to a unique point M', denoted as f(M) = M'. The concept builds upon prior knowledge from middle school, including point symmetry and line symmetry. The teacher explains that transformations are like a 'magic bag' containing various transformation rules, with translation being the first specific transformation to be studied in detail.

A symmetry group is a mathematical structure that captures all the ways an object can be transformed while preserving its essential properties, defined by four axioms: closure, associativity, identity, and inverses. These groups range from simple mirror symmetry (order 2) to complex 3D crystallographic symmetries (230 space groups), with applications in understanding molecular structures, crystal formations, and even aircraft navigation through Euler angles.

Symmetry is defined as a mapping preserving an object's properties, with identity always being a symmetry. Regular n-gons have 2n symmetries (n rotations + n reflections). Regular polyhedra symmetries are calculated by multiplying face count by face symmetries: tetrahedron (4×6=24), dodecahedron (12×10=120). A group has associative operation, neutral element, and inverses. Symmetry groups consist of all symmetries with composition as operation. In physics, space-time elements are events; symmetries preserve physical laws. Galilean transformations work for Newtonian mechanics but fail for electromagnetism, requiring Lorentz transformations in special relativity. Noether's theorem links symmetries to conservation laws: translations preserve momentum, time translations preserve energy, rotations preserve angular momentum.

Symmetry groups are mathematical structures that describe the set of all transformations (rotations and reflections) that map a geometric figure onto itself. For planar figures, any finite symmetry group is either a cyclic group Cₙ (containing n rotational symmetries) or a dihedral group Dₙ (containing 2n symmetries: n rotations and n reflections). For example, a square has the symmetry group D₄ with 8 elements (identity, three rotations of 90°, 180°, 270°, and four reflections), while an equilateral triangle has D₃ with 6 elements. In three-dimensional space, finite rotational symmetry groups are limited to Cₙ, Dₙ, T (tetrahedral, 12 elements), O (octahedral, 24 elements), or I (icosahedral, 60 elements).

The symmetric group Sₙ is the group of all permutations (bijections) of a set with n elements, with composition of functions as the operation. The order of Sₙ is n! (n factorial). Permutations are often denoted using two-line notation. The symmetric group is fundamental in combinatorics and algebra, serving as a prototype for understanding more complex group structures. The symmetric group S₃ is isomorphic to the group of symmetries of an equilateral triangle, which includes: the identity, three rotations (0°, 120°, 240°), and three reflections (across the three axes of symmetry). The symmetry group of a regular n-gon is called the dihedral group Dₙ, which has 2n elements. For n=3, D₃ = S₃. For n=4 (square), D₄ has 8 elements. These groups are subgroups of Sₙ and capture all transformations that preserve the geometric object.

Symmetry groups are mathematical structures that classify particles and understand their interactions. A group is defined by four properties: closure (combining elements produces another element in the set), identity (an element that leaves others unchanged), inverse (every element has a counterpart that produces identity), and associativity (order of combining three elements doesn't matter). Groups can be commutative (abelian) or non-commutative. Key examples include U(n) for unitary transformations, SU(n) for special unitary transformations with determinant 1, O(n) for orthogonal matrices, and SO(3) for 3D rotations. These groups form the mathematical foundation for describing symmetries in particle physics.