The Complete Blintz Part 10: Generalizing the Blintz

The standard blintz is defined as being folded from a square, and it has a range of wondrous properties. What happens if we try other paper shapes? And what other generalizations can we find?

Properties of the Square Blintz

Literally, to blintz is to fold all corners to the center. However, for a square, there is a range of defining and derived properties that we might choose to generalize when using the blintz as a tool from our toolbox. They may generalize in different ways, as we saw in the Pentagonal Lemon Squeezer, a generalization of the Fortune Teller.

Thus, before we can generalize the blintz we need to study its properties.

Alignments

The square blintz crease pattern has three equivalent specifications.

In The Complete Blintz Part 7: Deconstructing the Corner Fold, we learned how to fold a single corner. The blintz directly inherits the corner landmarks and additionally may use the first corner as a landmark for folding the remaining corners:

Center alignment: Fold the corners to the center.
Edge alignment: Fold between the edge midpoints.
Median alignment: Align the edges with the medians.
Corner alignment: Use the first corner fold as a landmark for the rest.

The first is the standard definition of a blintz and might be perceived as the intention. That the others are equivalent for the square but behave differently with other paper shapes must be considered when we want to transfer a crease pattern to another shape.

The first corner serves as the landmark for the rest.

As for the last option, the first corner defines both the center, usable for the other three corner folds, and the medians along the two neighbor folds.

All Corners

Normally, the blintz folds all corners. Isao Honda even calls it the all corner fold.¹ However, there may be cases where it is more natural to fold only some corners. We might call that a partial blintz.

Corner Order

Unless you use the corner alignment, the blintz flaps may be folded in any order, independently of one another.

There are six orders in which to fold the flaps, three ways in two symmetrical cases each. Most commonly, the flaps are folded in rotational order, clockwise or counterclockwise, but you also see people taking the opposite flap second, or the neighbor flaps second.

Order of folding the flaps. Six distinct permutations, since for the first corner, we can select any of the four.

One might imagine generalizations — with more complex folding sequences — in which the freedom to choose folding order is lost.

Shape Preservation

Fold four corners to the center, rotate 45 degrees, and you get a new square.

Blintzing the square provides a new square,² rotated 45 degrees. The shape preservation is the very thing that makes blintzed bases from a square so interesting. Peter Engel describes the epiphany he experienced when he discovered this property and the connection with crease patterns:

“The blintz was the only widespread innovation in origami since the invention of the frog base. [...] It begins with a given geometric figure (the square), divides that figure in two (the four corner triangles make up half the area of the square), and produces a replica of the original. The resulting figure is, of course, self-similar. [...] I raced through combinations of folds [...] The blintz fold opened a floodgate of new techniques.”³

If the fibers of the paper have a pronounced direction or there is a directional decoration, we must take the rotation of the shape into account when planning the folding sequence.

Symmetry

The blintz is rotationally symmetrical at 90-degree steps and mirror-symmetrical in both diagonals and both medians.

For models such as Friedrich Fröbel’s Folds of Beauty, this is important as the symmetry forms the kaleidoscopic effect the beauty is based upon.

Folds of Beauty are highly symmetrical.

Number of Flaps

Every blintz adds at least four extra flaps. A square has four corners that may function as flaps. Blintz once, now you have eight. Blintz again, and you have 12.

Left: Blintzes alternate direction but preserve shape. Right: The Lotus Flower from six blintzes exhibits 24 flaps. Folded by Hans Dybkjær from one layer of a three-layer dinner napkin treated with methylcellulose.

Peter Engel states that blintzing will “double the number of potential flaps.”⁴ His underlying observation is that the number of modules doubles. A module is a small crease pattern that may be pasted together with other modules to form a larger crease pattern. As blintzing a base copies the creases of the inner square to the blintz flaps; that seems true.

Top left to bottom right: module, kite base, fish base, bird base, blintzed bird base, double-blintzed bird base, triple-blintzed bird base.

Indeed, it is captivating to see the progression of the modules in increasingly complex bases, forever doubling the number of modules.

Even the larger compound modules like the blintzed bird base begin to repeat themselves, interestingly overlapping each other.

The examples are deceptive. The module does not exactly produce the kite base, the fish base, the bird base, etc. More importantly, the number of modules only doubles if the modules fit inside the triangles of the blintz pattern.

First row: Module-generated patterns. Second row: Actual patterns of the kite base, the fish base and the bird base.

Consider the kite base.

The doubling conjecture fails as soon as the base is not composed of modules that fit into the triangular blintz pattern. There is no recognizable repetition over the four blintz flaps, only the single mirror symmetry of the original. The blintz does not add symmetries; it only copies what is in the base.

In reading order: Kite base crease pattern, blintzed kite base crease pattern, folded kite base, folded and blintzed kite base, rearranged blintzed kite base with free flaps.

The blintzed kite base has six flaps, four along the edge and two in the middle. If we blintz, we get 14 flaps, of which 12 are visible. We can rearrange the paper to release the two hidden flaps. So, clearly, the growth may be steeper than a doubling.

Layers

The square blintz provides perfect coverage of the paper. None of the flaps overlap; no part of the front side is left uncovered. That is, the blintz adds exactly a single layer, which makes the blintz a double-layered model.

As a consequence, the blintz is halving the area⁵ — and doubling the thickness, which is why the lotus flower seldom exhibits more than three or four blintzes.

We might also view it the opposite way: A square is grafted with four triangles, each one, a quarter of the original, doubling the area. And then the grafting, the corners, are folded away to halve the area again and double the thickness.

Properties Summary

In the sections above, we identified a number of properties of the blintz for a square:

Case: A numbering for reference.
Illustration: A crease pattern of the landmarks and the creases, and the folded result.
Alignment: Align each flap with the center, the two edge marks, the median or a previous corner.
Corners: Does the move use all corners (standard), or is it a partial blintz?
Order: Are the flaps free of each other, or do they depend on or interfere with each other?
Shape: Is the blintz shape the same as the paper, or is it new?
Layers: A list of the numbers of layers in the result.
Area: The proportion of the blintz area to the original — the standard blintz has an area of \(1:2\). The proportion may be given exactly, or as less than half (\(< 1:2\)) or greater than half (\(> 1:2\)).
Symmetry: The blintz symmetry can be none, uniaxial, rotational, diagonal, median, or all of these.

I have omitted the numbers of flaps, as this depends on the base and on shape preservation, the latter not being satisfied in most of the generalizations.

For reference, here is the complete table for square cases. They all provide the same end result. For the corner case, by definition the corners depend on each other.

Generalizing Polarity

The corner flaps may be folded on top or behind, and we will call that the polarity. The standard blintz takes all flaps to the front. Technically, the definition states that to blintz is to fold the corners to the center, but it says nothing about the flaps being on the front, let alone on the same side.

Omitting rotational, front/back and left/right mirror cases, there are three extra cases where one or two flaps are folded to the back. See the table, presented by the order of mountain folds (m) and valley folds (v).

The traditional Sampan, a chinese boat, is an example of pairwise polarity (mvmv). The hoods at the ends of the boat and the color change are achieved through the blintz flaps.

Traditional Sampan has color change by reversing two opposite flaps (polarity mvmv). Folded by Hans Dybkjær.

Kunihiko Kasahara uses neighbor polarity (mmvv) to achieve the color changes of a Panda.⁶ Michael G. LaFosse uses neighbor polarity for a West Indian Manatee.⁷ The two outside flaps become feet, whereas the two inside flaps are unused and become unfolded, merely serving as landmarks.

Kasahara has a Wild Boar as well with three flaps on the inside and one of the outside (mvvv) to achieve the outside head and the suggestive big teeth.⁸ The three other flaps remain hidden. This is a blintzed square base with a mix of inside and outside blintz folds. Note: Gilad Aharoni⁹ translates it as wild boar, and my wife immediately recognised it as such. Personally I thought it was a badger.

Models by Kunihiko Kasahara. Left: Panda.⁶ Right: Badger.⁸ Folded by Hans Dybkjær.

The polarity options in the table above are seen from an isolated blintz view. If we view it in context, with the later folds, the standard blintz comes in two polarities: fold all corners to the front (vvvv), and fold all behind (mmmm). That corresponds to blintzed bases where we distinguish between an inside blintzed base and an outside blintzed base. See the discussion in The Complete Blintz Part 5: Creating via Blintzing.

An iso-area model is one where the two sides of the paper are displayed equally.¹⁰ The square blintz is iso-area when folded with either of the 2-2 polarities, mvmv or mmvv.

Generalizing How the Four Corners Get to the Center

Getting the corners to the center, from a cupboard fold base and from a kite base. Folded by Hans Dybkjær.

Is it a blintz whenever you get all four corners to the center? The examples above certainly do not feel like blintz folds. But the properties have much in common with the blintz, and they are generalization candidates.

The base of the cupboard fold blintz is shape-preserving, so blintzed bases can be made from it. With double the layers, it is, however, inefficient, and it is highly order dependent, interlocking much of the paper. Two reasons why the standard blintz is so useful are its free, independent flaps and that it consumes exactly one uniform layer of paper.

Both might be starting points for genuine origami models. The book fold blintz opens up into a cleaner variant of the traditional catamaran.

Likewise, an alternative salt cellar may be made from the book-fold-blintzed square base.

Catamaran from a book fold blintz and an alternative salt cellar from a book-fold-blintzed square base. Folds by Hans Dybkjær.

It is left as an exercise for the reader to reproduce these models and to find other uses of the book fold blintz as well as practical uses of the kite blintz.

Generalizing Folding Sequence Order

We have seen that the corners might be folded to the center in completely different bases. A smaller generalization may be to look at the order in the folding sequence. Is it still a blintz if you fold three corners at first, then do another 10 unrelated folds, and then fold the fourth corner to the center?

Let us look at some real-world examples.

Median Blintz

Blintzing via the median.

Creases may be difficult to discern when you need them as landmarks, for example due to lighting or coloration, whereas a folded edge is easy to fold against. A common method to counter this when blintzing is to fold a median, fold the corners pairwise to the folded edge and finally unfold the median.

This is only a slight generalization, and it is clearly a blintz, as the final step displays the fully completed blintz.

Folds of Beauty

Fröbelian origami had Folds of Beauty that were often folded from what is today sometimes called the windmill base. In Fröbel’s time, and often today, it is made by double blintzing as for the Fortune Teller and then rearranging the layers.¹¹

Folds of Beauty in the Fröbelian tradition.¹²

The creative aspects of the Folds of Beauty have been rediscovered by origamists such as Kunihiko Kasahara¹³ and Leyla Torres, who nicely demonstrates the creative aspects on her blog page on windmill folds.¹⁴ Note how she makes only one blintz. The second is converted into two cupboard folds thus avoiding reversing creases when collapsing.

In Germany, materials for Fröbelian folding are still in use and are sold through Netzwerk Lernen, for example.¹⁵ They use a folding sequence that does not explicitly include a blintz, but they achieve the same end result through other means.

Scottish Dog

Erik Rønholt¹⁶ first folds and unfolds three corners. Then he folds the final corner in and lets it stay there.

Later in the model, all the blintz creases become refolded in the valley folds of the original creases.

Blintz in Erik Rønholt’s “Nemme foldede dyr.”

This is not a blintz, even if we go through all the motions, as the full blintz is never present at once.

Masu Box

What if you never make an explicit blintz and yet end up with all four corners at the center? Isn't that a blintz with a circumspect folding sequence?

Surely the Masu Box starts with a blintz? Most instructions do start with a blintz, only differing in how carefully they create the landmarks from the medians and/or diagonals. And after that, we fold and unfold sides to center, unfold, and collapse, just as we saw in The Complete Blintz Part 8: A longer Masu Box by Grafting.

But no, some origamists like Louise Svendsen¹⁷ and Isao Honda¹⁸ use a different sequence with no blintz.

Instead, Svendsen folds only two corners to the center then proceeds to fold the long sides. Her sequence avoids two unfoldings, folding over the closed blintz edges and reversing part of the crease of the two end-blintz flaps. However, step 6 is a very thick fold and introduces two creases of the wrong polarity for the final collapse, in contrast with the usual sequence.

This is not a blintz, as the full blintz is never present at once, even if the four corners end up in the center, an alluring blintz look-alike. This is emphasized by the crease pattern, which reveals that the blintz pattern is only intact along two of the sides, the other two blintz lines becoming split into mountain and valley segments.

No blintz in Svendsen’s version of the Masu box.¹⁹

Finishing the masu box,²⁰ and crease pattern of the masu box.

No Blintz in the Waterbomb?

Two almost identical crease patterns, were it not for the mountain-valley assignment. Left: standard outside blintzed square base. Right: step toward the waterbomb.

The standard way of blintzing is to blintz first, then fold whatever is needed. However, in folding the waterbomb there is a step where all four corners are folded to the paper’s center as well and which superficially looks exactly like the blintzed square base.

The difference is that in one, we fold all corners to the center first, then fold the square base, while in the other, we fold the triangle base first, then fold the (paper’s) corners to the (paper’s) center.

However, the waterbomb step is not a blintz.

The valley-mountain assignments are completely different.
The step transforms a triangle into a square: It is not shape-preserving.
The step does not operate on all the visible corners of the flat shape. Rather, it takes two twin corners and leaves the third corner intact (the third corner being the center of the original paper).
The step does not take the corners of the shape to the center.

In particular, the latter two properties seem to be contrary to the intention of a blintz.

Try to fold the two crease patterns above. You will realize that the standard blintzed square base is completely open at the bottom point, whereas the pseudo-blintzed triangle is locked at the bottom, which leads to the closed waterbomb.

Waterbomb base via blintzing through the square base.

We may go through the blintz to produce the waterbomb. The sequence is exactly how blintzed bases are used: First blintz, then fold the base, and finally pull out and use the blintz flaps as needed. In this case, though, it is totally inefficient.

Shape Generalizations

Regular Triangles

Triangle reference points.

The regular triangle blintz reference points are just as symmetrical as those of the square. Nevertheless, the center, edge and median references yield different results.

The regular triangle center blintz (case a) is iso-area on one side; it is not double-layered, but a mix of single and double layers. The edge-mark blintz (case b) achieves a whopping four layers, but is actually shape-preserving.

The median alignment divides the angle divider into \(1\) and \(a = \sqrt3 - 1 \).

The example in case c interlocks and is interesting because the corners fit snugly inside the pockets in that each corner touches the angle divider of the next corner fold. To see this, observe that:

The height \(h\) divides the triangle into two 30-60-90 triangles which have sides \(1\), \(2\), and \(h=\sqrt3\).
By construction, the crease lines aligning the corners with the median lines bisect the 90 degree angle.
Hence we know the angles \(A=\pi/4\), \(B=\pi/6\), and \(C=\pi - \pi/4 - \pi/6\).
Using the trigonometric relations we get: \[a = c \frac{\sin{A}}{\sin{C}} = \frac{\sin{(\pi/4)}}{\sin{(\pi-\pi/4-pi/6)}} = \sqrt3 - 1 \]
Hence \(HX = HC = 1 \).
Since by construction \(HX\) is aligned with \(h\), the point \(X\) will end up at \(C\), QED.

Silver Rectangles

Rectangle reference points.

The silver rectangle has proportions \(1 : \sqrt2\), and in the form of A4 copy paper it is likely the most widespread paper format in the world.

Rectangles are irregular, and the edge marks, center mark and median alignment cases no longer coincide. Even the different medians will provide different results.

The A-format median folds include areas \(A\) and \(B\), which are the triangles formed by folding corners along the short and long medians, respectively. Their intersection areas are \(X\) and \(Y\).

As for the areas of the median cases, we observe that:

Given the paper is \(1\) times \(\sqrt2\), the area is \(\sqrt2\).
Area \(A = \frac14\).
Area \(B = \frac18\).
The intersection of \(A\) and \(A\) is \(X = \frac{3-2\sqrt2}{4}\).
The intersection of \(A\) and \(B\) is \(Y = \frac{3-2\sqrt2}{16}\).

Yoshimoto Cube No. 3. Origami designed and folded by Hans Dybkjær. Plastic modules, Efalin connectors; compare to case b in the Rectangle table above.

The edge mark case (b) has been used for module connectors in an origami rendering of the famous Yoshimoto Cube no. 3, a transforming toy designed by Naoki Yoshimoto in 1971. Due to the plastic modules, the colored Efalin paper connectors are visible and decorative.²¹

For the median cases (c-i), each corner may align with two different medians. As the two medians of a rectangle are asymmetrical, we get \(2^4\) cases. However, a number of those are symmetrical, so we end up with the seven cases listed.

Most of these are not really interesting, but case (c) is the first step of the traditional banger. Also, it appears in a generalization of the traditional masu box, as we saw in The Complete Blintz Part 8: A Longer Masu Box by Grafting.

A longer masu box and its crease pattern. See also The Complete Blintz Part 8: A Longer Masu Box by Grafting.

The case (i) occurs as an envelope model, the Foldelope by Andrew Stoker.²²

Foldelope: blintz and final model. Designed by Andrew Stoker; folded by Hans Dybkjær.

Regular Pentagons

Pentagon reference points.

Blintzed pentagons are similar to the previous examples. The center, edge and median references yield different results. There are more cases of the median alignments, but they are not really interesting.

Note the first two cases: Fold corners to center and fold corners between edge midpoints. Using the two in combination, and going to opposite sides, as in the Fortune Teller, makes a very nice pentagonal Fortune Teller. This version should rightly be the usual Fortune Teller, as the five pockets exactly fit the fingers of one hand. Try it — it is a fun, or frustrating, exercise in finger dexterity. The standard square Fortune Teller is also known as the Salt Cellar.

Pentagonal Lemon Squeezer derived from the Fortune Teller.

Ligia Montoya designed flowers from pentagons and used the edge variation of the blintz for one of them. Laura Rozenberg has published a diagram.²⁴

Flower 4, designed by Ligia Montoya. Folded and photographed by Hans Dybkjær.

Regular Hexagons

Hexagon reference points.

Blintzing hexagons is somewhat similar to blintzing triangles, and we have a lot more symmetry than with the pentagons since both the diagonals and the medians mirror the shape. The center, edge and median references yield different results. However, there is a great variation of center folds.

Case (a) is of interest because it is one of the steps in the gorgeous Snowflake by Dennis Walker.²⁵ Ligia Montoya designed a very nice Flower from the same base; the diagram is in a must-have book by Nick Robinson.²⁶

Left: Snowflake. Designed by Dennis Walker. Folded and photographed by Hans Dybkjær. Right: Flower. Designed by Ligia Montoya. Folded and photographed by Hans Dybkjær.

Cases (e, f, g) are semi-blintzes: Only some of the corners are folded to the center. The first of those is interesting as it is double-layered (no overlap) like the real blintz.

Ligia Montoya’s Flower #3 (below) uses the first semi-blintz, where every second corner is folded to the center.²⁷ It is obviously a doodle fold: The folding sequence feels like playing around with the geometry; the design misses an opportunity to use color change by simply avoiding the turning over before reversing the petals; and most importantly, the semi-blintz is unused, so the design can be folded from a triangle with no modifications to the folding sequence. So essentially, this use of the semi-blintz manages only to convert duo paper into mono paper.

Variations of Flower #3 by Ligia Montoya. Left to right: Original, color-changed variation, triangle variation. Folds, photos and design variations by Hans Dybkjær.

The Circle

Circles have no corners.

A circle has no corners and no straight edges, merely one round edge. Hence there are no corners to fold in and no obvious all-corner folds. So how do we blintz a circle?

Two answers are to inscribe the circle in a polygon, which provides edge landmarks, or to circumscribe the circle around a polygon, which provides corner landmarks.

As an example, let us consider how to square the circle and use the square reference points to blintz the circle. We can do this in two ways: by inscribing the circle or by inscribing the square.

Circle blintz models. Left: From inscribed square. Right: From circumscribed square.

In the inscribed circle case, the circle is smaller than the square, and the paper does not reach the center. In effect, we use the edge marks of the square as reference points.

In the circumscribed square case, the circle is larger than the square. The corners will reach the center, but there is excess paper along the outside of the square that we must handle. We may dispose of that excess paper in many ways, for example by making rabbit ears. However, we have chosen to fold the excess paper behind before folding the square corners into the center.

The decorative cushion fold is made by blintzing a circle circumscribing a square.

Looking closely, you may see that this amounts to doing the inscribed circle blintz from the back side followed by a standard blintz from the front. So the first case is part of the second case. Also, we know this sequence from the Salt Cellar and the Spanish Box. In particular, the Spanish Box looks nice and exposes some of the roundness. A common problem with circular paper is that when folding, all visible edges quickly become straight.

Left: Salt Cellar from a circle. Right: Spanish Box from a circle. Designed and folded by Hans Dybkjær.

Endnotes

1. Isao Honda, The World of Origami (Tokyo: Japan Publications Trading Company, 1966. Second edition, 8th printing, 1972) 36, 75. Library of Congress catalog card 65-27101. [back]

2. T. Sundara Row, Geometric Exercises in Paper Folding (New York: Dover, 1905). ISBN 0-7893-1341-3. [back]

3. Peter Engel, Folding the Universe — Origami from Angelfish to Zen (New York: Vintage, Random House, 1989), 75-77. ISBN 0-394-75751-3. [back]

4. Engel, Folding the Universe, 75. [back]

5. Row, Geometric Exercises in Paper folding, 7. [back]

6. Kunihiko Kasahara, Origami Made Easy (Japan Publications, USA, 1973, 18th printing, 1988), 104. ISBN 0-8704-253-6. [back]

7. Michael G. Lafosse and Richard L. Alexander, Origami Art (Singapore: Tuttle, 2008) 77. ISBN 978-4-8053-0998-8. [back]

8. Kunihiko Kasahara, Joy of Origami (Japan, Sojusha, 1996), 88. ISBN 4-916096-32-0. [back]

9. Gilad Aharoni, review of Joy of Origami, by Kunihiko Kasahara, https://www.giladorigami.com/BO_Joy.html. [back]

10. Kunihiko Kasahara and Toshie Takahama, Origami for the Connoisseur (Tokyo: Japan Publications, 1987), 26. ISBN 0-87040-670-1. [back]

11. Marie Müller-Wunderlich, “Die Grundform zu den Schönheitsformen B,” in Die Fröbelschen Beschäftigungen, 2 Heft: Das Falten (Leipzig: Friedrich Brandstetter, c. 1900), 18. [back]

12. Müller-Wunderlich, “Grundform,” Tafel 10. [back]

13. Kunihiko Kasahara, “Beautiful Pattern Folds,” in The Art and Wonder of Origami (England: www.Apple-Press.com, 2004), 42. ISBN 1-84543-061-1. [back]

14. Leyla Torres, “Windmill Base Variations,” text and video, accessed March 13, 2022, https://www.origamispirit.com/2011/05/windmill-base-variations-video. [back]

15. Netzwerk Lernen, “Unterrichtsmaterialien” (Learning network, “Teaching materials”), accessed April 6, 2021, https://www.netzwerk-lernen.de/advanced_search_result.php?keywords=origami. [back]

16. Erik Rønholt, Nemme foldede dyr (Denmark: Forlaget Cornelia, 2003), 32. ISBN 87-985354-1-2. [back]

17. Louise Svendsen, Japansk papirfoldning (Denmark: Forlaget Notabene, 1972), 27-29. [back]

18. Honda, The World of Origami, 72. [back]

19. Leyla Torres, “Windmill Base Variations,” text and video, accessed March 13, 2022, https://www.origamispirit.com/2011/05/windmill-base-variations-video. [back]

20. Leyla Torres, “Windmill Base Variations,” text and video, accessed March 13, 2022, https://www.origamispirit.com/2011/05/windmill-base-variations-video. [back]

21. Hans Dybkjær, “Euler Tours and Origami Cycles,” Journal of Symmetry 28, no. 3 (2017): 321-333. [back]

22. Andrew Stoker and Sasha Williamson, Fantastic Folds (London: Phoebus Editions, 1997), 24. ISBN 0-297-83546-7. [back]

23. Robert Harbin, “Saltkar,” in Origami — Japansk papirfoldning (Copenhagen: Notabene, 1970), 28. ISBN 87-7490-007-2. [back]

24. Laura Rozenberg, “Flower 4,” in Paper Life — The Story of Ligia Montoya (New York: CreateSpace Independent Publishing Platform, 2016), E-Book. ISBN 978-1533312952. [back]

25. Hans Dybkjær and Laila Dybkjær, Origami — Foldefamiliens jul (Copenhagen: Papirfoldning.dk, 2010), 44. ISBN 9-788799-235827. [back]

26. Nick Robinson, The Encyclopedia of Origami (London: Search Press, 2004), 118. ISBN 1-84448-025-9. [back]

27. Rozenberg, “Flower 3,” in Paper Life. [back]