While you may already be familiar with origami terms like mountain and valley folds, sinks, squashes, swivels, and rabbit ears, tessellations have a vocabulary of their own.
Unfortunately, the vocabulary of tessellations is not standardized across all tessellation designers and each tessellation book uses different definitions - and different colors and symbols for their diagrams.
This can make it hard to learn, since everyone is saying things in different ways.
I try to stick to consistent definitions throughout my teaching, and to explain my terms in plain English - no math speak.
Here's my list of the most important terms to understand when you're starting to fold tessellations:
Grid: Pattern of evenly spaced folds that are used for referencing other folds in a tessellation
Square Grid: Grid folded in two perpendicular directions, typically on a square sheet of paper
Triangle Grid: Grid folded in three directions with 60 degree angles at each intersection, native to hexagonal paper and also folded on squares and rectangles
45-Degree Grid: Square grid with either half or all diagonals added
Grid (Square/Triangle): Smallest shape outlined by grid lines on a finished grid
Grid Spacing: The distance of one grid line between two neighboring grid lines in another grid direction
Single Diagonal 45-degree Grid: Square grid with one diagonal added through each grid square, forming a checkerboard pattern of alignment
Double Diagonal 45-degree Grid: Square grid with both diagonals added in every grid square
Mountain fold: Fold that forms a ridge on the side of the paper facing you - shown with red solid lines
Valley fold: Fold that forms a trough on the side of the paper facing you - shown with blue dashed lines
Pleat: A pair of parallel mountain and valley folds
Pleat Depth: The number of grid spacings apart that the mountain and valley folds of a pleat are located, typically one spacing
Tube Pleat: A pair of two parallel, mirrored pleats with two grid spacings between the closest folds of the two pleats
Accordion Pleats: Two or more pleats in the same orientation and direction that stack on top of each other in the folded form to make an accordion/fan shape
Pleat Overlap: Two (or more) non-parallel pleats that cross each other, with one pleat completely closed when the second pleat is folded
Twist: Rotationally symmetric interaction of at least 3 pleats, with an interior shape created that is not part of the pleat folds
Pleat Intersection: Interaction of multiple pleats that is not an overlap or a twist (Note: Ben Parker's definition includes pleat overlaps and twists in the category of pleat intersections)
Crease Pattern (CP): Diagram showing where the mountain and valley folds go on the paper to fold an origami pattern; for tessellations this commonly includes a grid in addition to the folds
Tiling: An abstraction of the structure of tessellations where each twist or pleat intersection gets its own box; used to categorize and design tessellations and study their symmetries
Tessellation: Repeating 2-dimensional pattern
Origami Tessellation: Repeating pattern folded from a single sheet of paper
Whew - I think that covers just about everything!
If you'd like to learn more about each of these terms, check out my Tessellation Foundations series where I show how each of these terms are put into practice while teaching the fundamental skills needed to start folding tessellations.