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Kellogg’s Variety Pack as peace-maker We ended our previous post with a 1959 commercial showing how Kellogg’s Variety Pack could satisfy a consumer with differing Jekyll-and-Hyde cereal tastes. A year earlier, in 1958 Leo Burnett (Kellogg’s advertising agency) created this “settles all differences” campaign. Here, they illustrate the ways in which Kellogg’s Variety Pack “settles […]
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Define: Unitary Packaging Lately I’ve been thinking about “unitary packaging.” Although, I can’t seem to find any definitive definition for that term. Some people use it to describe a multi-pack of smaller packages. A six-pack carton of beer or soda, for example. Or the two-pack milk cartons we were looking at in an earlier post. But people […]
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Not everything in my wallet relates to package design. But when I got this email last month from New York Public Library, about their Lou Reed library card I immediately wanted one. Special Edition Lou Reed Card Available at the Library for the Performing Arts To celebrate the opening of the Lou Reed Archive, The Library […]
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Left: our house, circa 1890s (photo via Collection of Historic Richmond Town); right: boxer Al Roberts in 1919 The Staten Island Adonis I’m no heavyweight, but this month (and last month) I’ve been pondering a formerly famous local boxer. This all started for me the morning I woke up with the idea that I should search for our address on […]
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Pugilistic Packaging Revisited As a package designer, I know something about boxes, but next to nothing about boxing. So really I have no business writing anything on the subject of prize-fighting. (Although, I did once post this vintage advertising postcard under the headline: Pugilistic Packaging.) Recently, however, I’ve become obsessed with a not-so-famous boxer from the […]
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Some (non-intersecting) “twin packs” In our previous post we showed you a hypothetical structure for a pair of intersecting milk cartons. Today we bring you a similar (albeit non-intersecting) idea: “twin pack milk cartons.” Culled from American grocery store ads, I found these images in newspapers published between 1952 and 1976. I’ve also seen some more […]
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I was hoping that “intersecting milk cartons” were already a thing. But, alas, no example seem to exist online. So, for the 5th and final day of “Polyhedral Milk Carton Week,” I had to make it myself. What are we looking at? My 3D animation showing the intersection of two gable-top milk cartons. They intersect in […]
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Space-Pak “specimen” For day 4 of “Polyhedral Milk Carton Week” we promised to bring you evidence of International Paper‘s collapsible milk carton. I was hoping to find some decent photos of their Space-Pak™ carton showing its side panel score lines. But the best I’ve been able to come up with is the magenta “specimen” (above) […]
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For day 3 of “Polyhedral Milk Carton Week,” let’s talk about this black and white photograph entitled, Wandering Stars.

Oliver Helfrich and Antje Peter’s collapsible milk cartons

Oliver Helfrich made these gable-topped polyhedral paper sculptures for the 2010 book entitled The Book of Paper. Sharing authorship of the image, of course, is Antje Peters, co-author of the book, who made the photographs.

From milk cartons to tissues and take away coffee cups, paper plays an integral role in our day to day lives and yet we often take it for granted.

They did mention “milk cartons” in the press release for the book. But nowhere can I find any further discussion of the Wandering Stars construction.

Wondering about the Wandering Stars title

Not sure why they gave these abstracted milk cartons that particular title. The ancient Greeks called planets “Wandering Stars” (ἀστήρ πλανήτης) and they named the galaxy after milk (γάλα, gala).

Maybe that’s the reason? But, then again, it might mean something else altogether.

We’ve seen lots of packaging patents for collapsible bottles and the like. Here, Helfrich has whimsically invented some geometrically-collapsible milk cartons. The varying heights suggest a sequence, although it’s open to interpretation whether these cartons are being collapsed or extended.

The thing is, International Paper did, in fact, produce a collapsible milk carton in the 1970s. (Something we will be looking into on day 4 of “Polyhedral Milk Carton Week.”)

 

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1. The non-developable ruled surface named “milk carton”

(Day 2 of “Polyhedral Milk carton Week”)

Just learned something new! Someone has actually named a certain mathematically-defined hyperbolic surface: “milk carton.”

I found out about this on the French website, mathcurve.com, an online ENCYCLOPÉDIE DES FORMES MATHÉMATIQUES REMARQUABLES.

Looking more deeply into this somewhat hyperbolic milk carton, however, I think that it was really named berlingot and then the website translated the word into English.

If you look up the definition of “berlingot” you find that it has two meanings. It can mean either a tetrahedral-shaped twisted hard candy, or a tetrahedral-shaped (Tetra-Pak) carton. And (in the real world) one of these definitions can sometimes contain the other.

The MathCurve Website also tells us that another possible name (not French) for the berlingot surface could be “humbug.” (Since “humbug” is another name for this kind of tetrahedral candy.)

2 more “milk cartons” after the fold…

also known as: “quartic with two double lines”

Cundy & Rollett mentioned humbug candies in their discussion of the surface in Mathematical Models, although they called the surface a “quartic with two double lines.”

A simple but not very convincing example of such a surface is provided by the ‘old-fashioned humbug’. These potent sweetmeats seem to be made by twisting the material in just this way between two perpendicular skew lines at their extremities.

I was confused, at first, by the way the constructors of these models (and diagrams) sometimes show perpendicular ellipses at the ends and other times just perpendicular straight lines. Then I noticed the red threads in the model below. Those red lines show us where the ruled lines all cross—the effective edges of the “milk carton” form. (It’s easier to see how surface intersects itself in this photo.)

Also significant, but not super-obvious: the “milk carton” has a circular cross section where x, y and z intersect in the diagram above.

package design and non-developable surfaces

Richard P. Baker’s model, “Model of a Quartic Scroll #84” (circa 1909) via: Smithsonian Learning Lab

So why do they call these ruled surfaces non-developable?

Because we can’t make them with flat sheets of paper. Unlike cones or cylinders, this type of form has “compound curvature.” (like a sphere) And being “non-developable” makes a shape impractical for use as a folding carton.

MathCurve even cautions us,

Be careful, an [actual] milk carton … made of paper is a developable surface, made from a tetrahedron template by bending the edges…

And yet they’ve named this non-developable shape after a milk carton. What gives?

According to one commenter on Math.StackExchange: what “gives” is the material, itself.

The curved nature of the real-world milk cartons is because the material is slightly extensible.

In the realm of mathematical milk cartons, however, there is no such stretching or compression permitted. And hence, a doubly-curved surface is considered “non-developable.”

2. A non-developable ruled surface made from a milk carton

As we saw with yesterday’s milk carton torus, some people are also using milk cartons to make their hyperbolic, non-developable surfaces.

Joel Cryer from Wisconsin put together a lesson plan entitled Saddle Geometry. Here, he demonstrates how he recycled a small almond milk Tetra-Pak as a supporting structure for a hyperbolic paraboloid model.

(See also his: Bottle Biology instructions)

3. Milk cartons bulge due to “creep deformations”

Things start to get confusing when you consider how real-world milk cartons tend to bulge. As we are remind by another milk carton lesson plan, “If you look at the full carton … you can see that it bulges so isn’t strictly a rectangular prism.”

So, on the one hand, you can only make a paperboard milk carton of a “developable” shape. But, on the other hand, your carton inevitably assumes a “non-developable” hyperbolic shape. (Mathematically, speaking.)

A milk carton’s tendency to bulge will sometimes grow into a full-fledged packaging problem.

Packages made from paper and plastics have many advantages compared to packages made from other materials, e.g. paper and plastics are easy to manufacture and are environment-friendly. A drawback is that beverage packages made of paper and plastic materials stored for a long time, deform due to creep deformations, causing swelling of the packages, which adopt the shape of a “balloon”, shown in Figure 1.1. This phenomena is called bulging. In the end the final customers, i.e. you and I, will think that the content is fermented and will pick another package. Thus, the swelled packages will be rejected and unsold.

ANALYSIS OF CREEP IN PAPERBOARD PACKAGES WITH PLASTIC TOPS
Master’s Dissertation by Jakob Arvidsson and Jesper Grönvall
Division of Structural Mechanics, LTH, Lund University, Sweden
(in cooperation with Tetra Pak R&D AB)

See also: Packaging form as product differentiator

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