Physics of Paper Engineering

The decomposition of the printed book underlying the operation of the movable book was used historically for different purposes (educational, mnemonic, ludic, cryptographic), .... The developmental stages of this type of book... were marked by the different paper-engineered devices used: starting from a two-dimensional structure, simulating movement and/or three-dimensionality, then came three-dimensional structures which broke the conventional structure of the book, going beyond the confines of the pages. 

- Mara Sarlatto, "Paper Engineers and Mechanical Devices of Movable Books of the 19th and 20th Centuries" Italian Journal of Library, Archives and Information Science, Vol. 7, n. 1 (January 2016). Art. #11610 p. 90 

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An illustration showing the parts of an out of the page pop up. The center fold of the base page is called a gutter fold. Folds raised toward the viewer are mountain folds and folds pushing back away from the viewer are valley folds.
The picture illustates a tip-on pop up element. Unlike the out-of-page pop-up that is created directly from the base page, a tip-on is a piece of paper added to the base paper to create the pop up. The points were the tip-on is connected to the base page is called a seam.
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Basic Principles

The basics of paper engineering rely on a foundation of math and science principles. The best way to understand the basic elements of paper engineering is observation through construction and manipulation. A few key concepts can clarify its mathematic and scientific foundations.  The first of which is Newton’s Third Law: 

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Newton's 3rd Law

For every action, there is an equal and opposite reaction.

“A force is a push or a pull that acts upon an object as a result of its interaction with another object. Forces result from interactions! .... Some forces result from contact interactions (normal, frictional, tensional, and applied forces are examples of contact forces) and other forces are the result of action-at-a-distance interactions (gravitational, electrical, and magnetic forces). According to Newton, whenever objects A and B interact with each other, they exert forces upon each other.... These two forces are called action and reaction forces and are the subject of Newton's third law of motion.” 

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Trigonometry

Trigonometry, the study of angles, is essential in the creation of successful angle folds and parallel folds. Whether or not the creator understand the math behind the structures, the pop-up will not work if the structure does not straddle the gutter fold and if the angles and lengths are not balanced. 

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Angle Folds (a.k.a. V-Folds)

Diagram showing an angle fold sitting on a base page. It has the angles highlighted in four different colors to illustrate that angles a & c equal angles b & d
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Angle Folds

Angle folds, also known as v-folds, may be right, acute, or obtuse angles. It is also possible to create asymmetric angles. The v-fold is created when the three creases of the pop-up element converge at the same point on the spine or gutter of the base. In each case, the angle on the pop up and the angle on the base relative to the left of the central fold of the base sheet must equal the angle on the pop up and the angle on the base relative to the right side of the central crease of the base. This is especially important when designing asymmetric angles. The angle of the pop-up element must be greater than or equal to the angle on the base in order for the pop up to "pop." 

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Diagram illustrating that the angle of the pop-up in relation to the gutter of the base page changes whether the pop-up will stand up straight, lean back, or lean forward.
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Parallel Folds

Parallel folds produce a wide range of shapes: boxes, tents, cylinders, cones, pyramids, boats, and arcs. Unlike the angle fold, the 2-3 main creases of the pop-up element are parallel to the gutter/spine. Regardless of the shape, the length of the element on Side A plus the distance of the element from the gutter on that same side must be equal to the length of the element on Side B plus the distance of the element from the gutter on the same side. This is true even if the element is asymmetric. The pop-up element must be larger than the distance between the base anchors. It should be noted that the parallelogram lies flat at 180° and only comes into dimension when held at less than 180° while the symmetric and asymmetric folds require the page be open to 180°.

Image illustrating parallel fold at 90 degrees and 180 degrees. The picture also illustrates that sides a and c equal the lengths of sides b and d