10.2 Three-Dimensional Figures, Transformations, Symmetry & Tessellations

Overview & Exam Relevance

This section continues Competency 004. It covers three-dimensional figures and their nets, rigid transformations, symmetry, tessellations, and transformations on the coordinate plane. These topics build spatial reasoning: the ability to picture figures, move them mentally, and see how parts relate to the whole.

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Three-Dimensional Solids, Nets & Euler's Formula

Three-dimensional figures are categorized into polyhedra (flat polygonal surfaces) and non-polyhedra (curved surfaces).

Prisms versus Pyramids

• Prisms: Polyhedra possessing two congruent, parallel polygonal bases connected by lateral faces that are parallelograms (rectangles in right prisms). Prisms are named by their base shape (e.g., triangular prism, rectangular prism, hexagonal prism).
• Pyramids: Polyhedra possessing one polygonal base connected to triangular lateral faces that converge at a single shared point called the apex.
• Non-Polyhedra: Curved 3D solids including cylinders (two parallel congruent circular bases), cones (one circular base converging to an apex), and spheres (set of all points equidistant from a central point).

Euler's Formula for Convex Polyhedra

For any convex polyhedron, the number of Faces ($F$), Vertices ($V$), and Edges ($E$) satisfies Euler's invariant formula: $F + V - E = 2$

2D Nets of 3D Solids

A net is a two-dimensional flat pattern of connected polygons that can be folded along edges to construct a complete three-dimensional solid without overlaps:

• Cube Nets: A cube consists of 6 square faces. There are exactly 11 unique valid nets that successfully fold into a cube. A straight strip of 5 squares with 1 extra cannot fold into a cube, nor can any arrangement where square flaps overlap.
• Instructional Utility: Unfolding solids into nets allows students to transition from abstract 3D spatial thinking to calculating total surface area by summing the individual areas of the 2D polygonal faces.

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Geometric Transformations & Symmetry

Transformations describe the movement or mapping of geometric figures on a plane.

Rigid Transformations (Isometries)

Rigid transformations preserve shape, angle measure, and side length; the pre-image and image are congruent:

1. Translation (Slide): Every point of the figure moves the exact same distance in the exact same direction. Coordinates change by adding or subtracting constants: $(x, y) \to (x + a, y + b)$.
2. Reflection (Flip): The figure is flipped across a fixed line of reflection, which serves as the perpendicular bisector between every point on the pre-image and its corresponding point on the image. Reflection reverses orientation (chirality).
3. Rotation (Turn): The figure is rotated around a fixed center of rotation by a specified angle and direction (clockwise or counterclockwise).

Symmetry

• Line Symmetry (Reflectional): A figure has line symmetry if it can be divided by a line into two halves that are identical mirror images.
• Rotational Symmetry: A figure has rotational symmetry if it can be rotated about its central point by an angle strictly less than $360^\circ$ and coincide exactly with its original outline. The order of symmetry is the number of times it matches within a $360^\circ$ rotation (e.g., a square has rotational symmetry of order 4, with an angle of rotation of $360^\circ / 4 = 90^\circ$).

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Tessellations

A tessellation covers a plane with repeated shapes and no gaps or overlaps.

• Regular tessellations use one regular polygon. Only equilateral triangles ($60^\circ$ angles), squares ($90^\circ$), and regular hexagons ($120^\circ$) work, because their angles divide evenly into $360^\circ$ at each vertex. A regular pentagon ($108^\circ$) cannot tessellate.
• Any triangle and any quadrilateral can tessellate when copies are rotated and translated.
• Semi-regular tessellations combine regular polygons, such as regular octagons and squares on a bathroom floor.
• Tessellations illustrate symmetry and transformations: translations, rotations, and reflections move a tile into every position. The artist M. C. Escher is famous for tessellations of birds and fish built by modifying polygons.

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Transformations on the Coordinate Plane

Translations, reflections, and rotations are rigid transformations: they preserve size and shape, so the image is congruent to the original. A dilation changes size but keeps angle measures, producing a similar figure. Elementary students begin with physical slides, flips, and turns of pattern blocks before using coordinates.