These creases form a planar subdivision of the paper called the **crease pattern**.

Mathematically, the paper is modeled as the interior region of a (typically convex) polygon. The crease pattern is given by the edges of a planar, piecewise-linear, (typically convex) tiling of the region. The figure to the right shows a crease pattern on a square piece of paper.

In a base, two faces incident on the same crease edge can remain flat, or can be folded. This
is formalized by the concept of **mountain** and **valley** folds. If the dihedral angle
between two faces is larger than pi, then the crease is said to be a mountain fold in the base.
If smaller, the crease is a valley fold. If equal, the crease is flat. This is illustrated in the
figure below.

Robert J. Lang further categorized two subsets of bases called *projectable*
and *uniaxial* in his paper "A Computational Algorithm for Origami Design." These
provide for a more strict form of base which Lang showed is amenable to an algorithmic
treatment.

In this figure the crease pattern is shown on the left, and a uniaxial base for the crease pattern is on the right. The orthogonal plane has been shifted downwards so that the shadow tree is clearly visible. The boundary of the paper has been colored blue in both figures, and the diagonal creases are shown in black while the horizontal and vertical creases are depicted with a dotted white line. The reader can clearly see that the boundary of the polygon is folded exactly onto the shadow tree.

All of the faces that project to the same arc of the shadow tree are called a *flap*.
The uniaxial base in the figure above has four flaps, each of which two faces of the base
project to. In a uniaxial base, the internal nodes of the shadow tree correspond to "hinge" creases that
are perpendicular to the orthogonal plane. Rotations of shadow tree arcs correspond to rotations
of the flaps around the internal nodes correspond to rotations of the flaps about the corresponding
hinge crease. The figure below shows two uniaxial bases for the same crease pattern as above. These
bases only differ by rotation around the hinge creases.

**Two uniaxial bases with the same crease pattern.**

The term *uniaxial* is used, because if the shadow tree arcs are aligned, as in the figure below,
the boundary of the paper is aligned along a single axis.

**If the shadow tree of a uniaxial base aligns, then the boundary of the paper aligns along a single axis.**

Lang formulated another version of the origami design problem, and solved it with his TreeMaker algorithm: given a desired shadow tree and polygonal sheet of paper, Lang's version of the problem is to produce a crease pattern for the paper such that there exists a uniaxial base whose shadow tree is the input tree.

**The origami design problem solved by TreeMaker.**

As we saw above, in a uniaxial base flaps can be rotated around hinge creases, which correspond to rotations of the corresponding shadow tree arcs around internal nodes. This allows us to generalize the geometric shadow tree to the concept of a topologically embedded metric tree. Given a shadow tree and a uniaxial base projecting to that shadow tree, a uniaxial base for any other shadow tree which is metrically and topologically equivalent can be reached by simple rotations of the flaps around their hinges. This reduces the problem of generating a crease pattern for a desired geometrically embedded tree to that of generating a crease pattern for a desired topologically embedded metric tree.

Furthermore, we can define a **standard alignment** of the tree by selecting a root vertex, and
"shaking out" the tree so that all other arcs fall downwards. All arcs are thus aligned along the
same line, with the topological ordering preserved. This alignment of the tree along a single
axes is the reason that the uniaxial bases are called "uniaxial." In the figure below we illustrate
the standard alignment (right) of a topologically embedded metric tree (left). Keep in mind that
the arcs in the standard alignment have been curved outwards for visualization purposes, but in
reality all arcs lie along the same line.

**A rooted topologically embedded metric tree (left) and its standard alignment (right).**

**A topologically embedded metric tree and two doubling cycles. The first doubling cycle is a Lang polygon.**

If a doubling cycle is a convex polygon such that for each pair of vertices corresponding to leaf
nodes, their euclidean distance is greater than or equal to the corresponding distance between the
leaves, then we call the doubling cycle a **Lang Polygon**. In the figure above, the first
doubling cycle is a Lang polygon, but the second is not (since by the triangle inequality, the distance
between vertices a1 and a2 is less than the distance in the tree between leaves a1 and a2).

In a Lang polygon we call vertices corresponding to leaf nodes **corner** vertices and
those corresponding to interior nodes **marker** vertices.

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