Acute V-Fold Tutorial: Compare Tilt and Mounting Angles

1. Measure the paper, rather than the photograph

If A, B, C, D and the page opening θ are still unfamiliar, start with the right-angle V-fold lesson.

Unfold the sheet and extend the mounting creases toward O at the lower end of the centre crease. The flat angles between the centre crease and these mounting lines are A and B.

With A=B<90° and matching mounting angles C=D, we have the symmetric acute V-fold discussed here. This lesson uses A=B=60°.

The mounting lines now slope while the upper edge stays straight. Classify the mechanism by its crease angles. A pointed outline or wings that look narrowly opened in a photograph do not establish that it is an acute V-fold.

2. Change only the paper angle

You can also follow the annotated diagrams below.

Keep the base fully open and compare A=B=90° with A=B=60°. Both examples use C=D=130° and the same mounting-edge and centre-crease lengths.

Watch the centre crease. The right-angle example stands perpendicular to the flat base, while the acute example leans toward one end of the spine. The paper angle changed; the book remains fully open and the camera stays in place.

Move the page opening to roughly 90° and repeat the comparison. The wings change pose during motion, so record the paper angle, mounting angle and current page opening together.

3. Change only the mounting angle

Keep the same A=B=60° sheet and change its mounting angles from C=D=130° to 150°. The wings, tabs and crease pattern all stay the same.

You can also follow the annotated diagrams below.

With the base fully open and the camera unchanged, the 150° example has a more upright centre crease than the 130° example. Only the direction in which the sheet attaches to the base has changed.

Measure C and D from the downward spine ray at O. The 150° mounting lines sit closer to the upward spine direction, yet their readings from the downward ray are larger. A different starting ray can reverse what someone means by increasing or decreasing an angle.

This comparison concerns these two valid mounting arrangements. Record the actual A, C and page opening rather than treating increasing C as a universal way to make every mounting direction more upright.

4. Why can some acute V-folds only open partway?

Select “C=D=80° · limited” in the mounting experiment. The paper angle remains 60° and the structure still moves through part of the range, but this model stops at roughly 123° rather than showing an unreachable 180° pose.

This value comes from the lesson’s ideal geometric model. A structure that moves and rises does not necessarily open all the way to 180°. If a physical sample resists, reopen and inspect it. Bending the paper under force does not establish correct mounting.

For the symmetric, non-degenerate acute configuration discussed here, a fully flat base requires C<A or C>180°−A, with A=B and C=D.

The equalities are degenerate boundaries and are not used for this exercise. Angle conditions do not replace checks for tab clearance, fitting inside the page or physical test folds. Make the 130° and 150° versions first.

5. Make two samples to compare the mounting angles

Download the C=D=130° practice pattern and the C=D=150° practice pattern. Their pop-up sheets match; their base glue positions differ.

Print landscape on A4 at actual size or 100%, with Fit to page disabled. Check the 20 mm calibration line first. Each base is 210×148 mm. The centre crease and both mounting edges are 45 mm long, and each tab is 24×8 mm. The tabs stop short of O.

1. Cut two sets of parts

Lightly score the dashed lines, then cut the black outer outlines. Red is the centre crease, green marks the mounting creases and grey marks the base spine. Keep both tabs and remove the surrounding gaps. Do not cut the dashed lines.

Compare the two pop-up sheets to confirm matching creases and tabs. Set each beside its base and check whether the base title says 130° or 150°.

2. Pre-fold and position without glue

From the marked face, mountain-fold the red and green creases and valley-fold the grey spine. Align the lower end of the centre crease with O and match the tabs to the yellow L and R areas. The tab faces marked GLUE face the base.

Fold the wings together and gently close the base. Check that each tab stays on its own page, the creases move and the sheet fits inside. The tabs do not need to meet at O.

3. Assemble at the matching glue positions

Glue one tab with a thin coat. Apply adhesive to the other, bring the mechanism into its pre-folded closed position and close the base to help locate it. Follow the adhesive instructions for pressing and setting, keeping glue within the tab areas.

Use the second set for the other mounting angle. Separate samples allow a side-by-side comparison at equal page openings while preserving the first assembly.

4. Let the bonds set, then compare equal states

Open both samples fully and compare their centre creases. Then open both to roughly 90° and observe how they fold away. Record the differences and slowly close and reopen them.

Check that both samples have clear creases and secure tabs. If either catches, locate the problem before adjusting it.

6. Classify the creases even when the silhouette changes

Once a basic sample opens smoothly, trim a small upper corner from one wing while preserving the centre crease, mounting creases and tabs. Repeat the same motion. Distinguish changes to the silhouette from changes to the creases that govern movement.

Finished works may suggest pointed forms, bodies or mouths. A pointed upper edge alone does not identify a pointed V-fold; check the creases and connections as well. First understand the ordinary acute configuration. Pointed forms, counter-folds and combinations can follow as separate topics.

7. Restore the comparison conditions before troubleshooting

To compare in the workbench, create a project from the basic V-fold element, place it on the spine and select it. Set the paper angle A to 60° and mounting angle C to 130°, then open the pages to 180°. Keep the camera fixed and change only C to 150°. The default triangular wings demonstrate the same angle relationships; use the downloads for this lesson’s straight upper edge.

Finally, identify the angle that makes this an acute V-fold. Explain why two A=60° samples can lean differently, and why movement alone does not prove a full opening range. The next lesson will apply the same observation method to obtuse V-folds.