Kamifusen (紙風船), meaning paper (kami) balloon (fusen), is a traditional Japanese toy. The balloon is constructed from semitransparent glassine segments put together in a manner similar to the way plastic segments make a beach ball. Typically 10–20 cm in diameter, the kamifusen is inflated through a small hole 8 mm or so wide and gently bounced on the palm of the hand, as illustrated in figure 1. Because it is made of paper, the kamifusen is light enough for indoor play yet heavy enough that keeping it aloft is a challenge.

Figure 1. Bouncing a kamifusen. The balloon’s hole is visible in the silver patch where the tapered ends of the different-colored wedge-shaped paper converge.

Figure 1. Bouncing a kamifusen. The balloon’s hole is visible in the silver patch where the tapered ends of the different-colored wedge-shaped paper converge.

Close modal

Kamifusen became popular in the early 1890s, but their origin is not clear. They used to be widely available at neighborhood snack shops called dagashiya (駄菓子屋) that sold inexpensive treats and toys for children. Nowadays such stores have become few and far between, but kamifusen can still be found as folk toys at souvenir shops and other specialty stores in Japan and elsewhere. Modern kamifusen also come in fruit or animal shapes or with printed patterns; some are more for display than for play. With its graceful bounce and pleasant sound of crumpling paper, the kamifusen continues to be a cherished and timeless toy. It is also an enigma.

Despite its hole being open to air, a kamifusen remains inflated. One might expect that bouncing the kamifusen would force air out of its hole and cause the balloon to deflate. Instead, the batting action actually increases the kamifusen’s degree of inflation. In fact, repeated bouncing can make a nearly deflated kamifusen swell to its fully inflated condition. Part of the kamifusen’s charm is its ability to be pumped up in that way—and the activity itself can be a lot of fun.

The counterintuitive behavior of the kamifusen’s inflation has previously been attributed by some physicists to viscoelasticity, the property that causes a crumpled sheet of paper to slowly unfold. Viscoelasticity can indeed expand a scrunched paper balloon. However, it does not fully explain the kamifusen’s inflation, since viscoelastic unfolding does not completely smooth out plain crumpled paper even if the paper is bounced like a kamifusen. Other physicists have suggested that the bouncing causes pressure changes inside the balloon that alternately push out and pull in air. That’s true, but it is not obvious how the pressure changes and why more air comes in than goes out—a requirement for the balloon to inflate. In this Quick Study, I explain the self-inflation of the kamifusen by taking a closer look at the pressure variations that occur as the balloon is batted about. I also touch on the material properties of the paper from which the balloons are made.

The puzzle of the kamifusen’s inflation is about airflow between the balloon and its surroundings. Air moves from whichever region—balloon or atmosphere—has higher pressure to the lower-pressure region, so I first focus on pressure.

Before being bounced, the kamifusen is in equilibrium with its surroundings and its pressure equals the pressure, p0, of the ambient atmosphere. But when batted by hand, the balloon deforms, which changes its pressure. The question is, how and in what way does that pressure change happen?

Because the kamifusen’s hole is small, the amount of air that can flow through it is limited. Thus, to first approximation, the total amount of air in the balloon is constant in time. When the balloon is struck, it suffers some plastic deformation (it does not fully return to its original shape after force is removed), but it also contracts and expands elastically, as illustrated in figure 2a; during that process pressure deviates from the ambient value. The elastic oscillation propagates to other regions of the balloon as sound waves analogous to the seismic waves that communicate an earthquake to the rest of the globe. As the waves propagate, pressure fluctuations force a small amount of air through the hole. When the balloon’s pressure at the hole is higher than atmospheric pressure, air is forced out; when it is lower, air is sucked in.

Figure 2. An oscillating kamifusen. When bounced, (a) a kamifusen oscillates between states that are contracted (red) and expanded (green) relative to the balloon’s equilibrium volume (black). Pressure changes accompany the oscillation. When the balloon is contracted, the inside pressure is higher than atmospheric pressure p0; when expanded, it is lower than p0. The pressure difference between balloon and atmosphere works to restore the balloon to its equilibrium position. As described in the text, the fluctuations propagate as sound waves. (b) Compared here are the average duration t and average pressure of the waves’ high-pressure (H) and low-pressure (L) phases at an arbitrary location in the balloon. As explained in the text, tH is shorter than tL, but pHp0 is greater than p0pL. Indeed, (pHp0 )tH = (p0pL )tL; in other words, the rectangles of width tH and tL have the same area.

Figure 2. An oscillating kamifusen. When bounced, (a) a kamifusen oscillates between states that are contracted (red) and expanded (green) relative to the balloon’s equilibrium volume (black). Pressure changes accompany the oscillation. When the balloon is contracted, the inside pressure is higher than atmospheric pressure p0; when expanded, it is lower than p0. The pressure difference between balloon and atmosphere works to restore the balloon to its equilibrium position. As described in the text, the fluctuations propagate as sound waves. (b) Compared here are the average duration t and average pressure of the waves’ high-pressure (H) and low-pressure (L) phases at an arbitrary location in the balloon. As explained in the text, tH is shorter than tL, but pHp0 is greater than p0pL. Indeed, (pHp0 )tH = (p0pL )tL; in other words, the rectangles of width tH and tL have the same area.

Close modal

If that pressure variation were symmetric between its highs and lows, the amounts of air flowing in and out of the kamifusen would be equal and the balloon’s volume would not change. Such, however, is not the case. When the balloon is struck, pressure increases simultaneously across the region where the hand impacts the kamifusen. But subsequent waves reflect and scatter off different parts of the balloon’s surface, which leads to a spread in the duration and amplitude of the pressure variation. The result, as illustrated in figure 2b, is an asymmetry between the high- and low-pressure states.

Specifically, at any given location inside the balloon, the average duration tH of the high-pressure state is shorter than the average duration tL of the low-pressure state. On the other hand, the average high pressure pH deviates more from atmospheric pressure than does the average low pressure pL. In fact, pressure difference and duration are inversely proportional to each other and satisfy (pHp0)tH = (p0pL)tL. The inverse relation is a statement of energy conservation; it equates the energy transported by the sound waves’ high- and low-pressure phases. The difference between kamifusen pressure and p0 is proportional to the power carried by the wave. And that power, multiplied by time, gives the energy transported.

Now that I’ve described how pressure changes, I turn to how the pressure variations affect airflow. The amount of air moving through the balloon’s hole depends on the speed and duration of the flow. Bernoulli’s principle, which relates fluid motion and pressure, tells us that the speed is proportional to the square root of the pressure difference. So the ratio of the flow speeds in the high- and low-pressure phases in figure 2b is not as great as the ratio of their pressure differences. But I’ve just shown that the duration of the two phases is inversely related to those pressure differences. So, for example, if pHp0 = 4(p0pL), the expelled flow has twice the speed but only ¼ the duration of the incoming flow. As a result, the amount of air flowing into the balloon is greater than the amount flowing out, and the kamifusen inflates.

Part of the kamifusen’s genius is the paper from which it is made. The paper is not only lightweight and relatively impermeable to air, but it also has a degree of plasticity that allows it to deform easily and retain its resulting shape. Because of those properties, the kamifusen inflates to a volume commensurate with its air content and maintains that volume until additional air is added. As a result, a squashed kamifusen can accumulate air and eventually inflate to its full size from repeated bouncing, even though the net pumping from a single bounce may be small. A balloon made of plastic, rubber, or any other material that does not share the key properties of kamifusen paper would not inflate as the Japanese balloon does.

Elastic waves, fluid motion, and the paper’s plasticity work together in the self-inflation of the kamifusen. The balloon’s deceptively simple design conceals an intricate process at work and attests to the ingenuity of the artisans who devised this elegant, intriguing toy.

Ichiro Fukumori

Jet Propulsion Laboratory, California Institute of Technology

In the January 2017 Quick Study “Kamifusen, the self-inflating Japanese paper balloon,” I explained why the balloon inflates when you bat it about with the palm of your hand. In brief, the kamifusen (紙風船) oscillates elastically, and during that process it spends a short amount of time in a high-pressure state in which air is expelled and a relatively longer period of time in a low-pressure state in which it sucks air in. Bernoulli’s principle guarantees that more air is sucked in than is expelled out. Detailed testing of the mechanism I described would require measurements of volume, pressure, airflow through the hole of the balloon, and the stress and strain of its paper skin. Nonetheless, some simple experiments can shed light on the mechanism I described.

Consider, for instance, a paper balloon in the shape of a cube instead of a sphere—a kakufusen (角風船). Traveling pharmacists called baiyaku-san (売薬さん) used to give out kakufusen (kaku means “angled”) as promotional items with ads printed on their sides, but nowadays kakufusen are less common than the round kamifusen. Like their spherical cousins, the kakufusen inflate by themselves when bounced on your palm. In particular, they inflate into convex forms with bulging faces and the maximum volume allowed by their construction. Because the fully self-inflated kakufusen is not cubic, the inflation is evidently not driven by the paper’s elastic tendency to return to its flat state. The bulging kakufusen is, however, consistent with the mechanism discussed in the Quick Study.

Further light on the nature of its inflationary mechanism comes from examining means that defeat the kamifusen’s inflation. For instance, poking an inflated kamifusen with a finger instead of bouncing it on a palm causes the balloon to deflate. The reason is that the small-scale finger poke produces a local plastic deformation that squirts air one way—out of the balloon—rather than an effective compression that generates the elastic oscillation fundamental to the balloon’s inflation. This poke response may also explain why most kamifusen are the size of a hand: Anything bigger would require something larger than a hand to keep it inflated and thus would not be well-suited as a toy.

A kamifusen also fails to inflate if its hole is too large, if it has too many holes, or if its paper skin is torn too much. In all those cases, too much air escapes from the balloon and an effective elastic oscillation never takes place. Kamifusen don’t inflate when they are softly caught and gently tossed instead of being bounced. That phenomenon, too, is consistent with the Quick Study’s explanation, in which the sudden impact of the bounce is essential to the balloon’s inflation. In general, kamifusen inflate more readily when they are bounced more sharply, provided, of course, that they do not rupture. When strongly batted, kamifusen also exhibit a jerky flight motion suggestive of air squirted in and out of the balloon due its elastic oscillation. According to the mechanism proposed in the Quick Study, a kamifusen would fail to inflate if it were bounced in a vacuum, because air fluctuation drives the inflation, not elastic rebounding of the balloon’s paper skin.

The mechanism described in the Quick Study implies that a kamifusen would deflate if it could be bounced from within or, equivalently, if it were to be suddenly expanded without a change in the quantity of air it contains. In that case, the low-pressure state would be the shorter-lived one, and Bernoulli’s principle would demand that more air be expelled than sucked in.

A sudden-expansion experiment would be an effective test of the ideas I proposed in the Quick Study. For now I leave that as a topic for future investigation. Plenty of room remains for a more rigorous analysis of the balloon’s behavior both experimentally and theoretically. Whether phenomena analogous to the kamifusen’s self-inflation exist elsewhere, particularly in the natural world, warrants examination. So do possible applications of the kamifusen mechanism.

Ichiro Fukumori

Jet Propulsion Laboratory, California Institute of Technology

In the January 2017 Quick Study “Kamifusen, the self-inflating Japanese paper balloon,” I observed that when you bat a kamifusen (紙風船) with your hand, the balloon oscillates between a shorter-lived, air-expelling, high-pressure state (average pressure pH) and a longer-lived, air-inspiring, low-pressure state (average pressure pL). Here I’ll present the algebraic details that determine the net airflow into the balloon.

Energy conservation demands that the oscillation’s high-pressure state has the same amount of energy as its low-pressure state. Equivalently, the high- and low-pressure phases of the oscillation’s waves carry equal amounts of energy. The product of energy density, duration, and wave speed gives the waves’ net energy transport per unit area at the kamifusen’s surface where the hole is. The difference between the wave’s pressure and atmospheric pressure p0 represents the energy density of the wave. The wave speeds in the two phases are the same because sound waves are non-dispersive. Therefore the equivalence of high- and low-phase energy transport is given by

pH-p0tH=p0-pLtL
(1)

where tH and tL are the average durations of the waves’ high- and low-pressure phases respectively. Equation (1) is equivalent to the statement that wave action (energy density times characteristic time scale) is invariant.

The relationship between the speed of air (vi; the index i is a stand-in for either H for the wave’s high-pressure state or L for its low-pressure state) flowing through the balloon’s hole and the difference between inside pressure (pi) and outside pressure (p0) is, from Bernoulli’s principle,

ρvi2/2=|pip0|
(2)

where ρ is the density of air. The change in density as the kamifusen oscillates is relatively small, so ρ can be treated as constant.

The net volume of air Q flowing through a kamifusen hole of area S is given by

Q=(vLtLvHtH)S.
(3)

In obtaining that equation I defined Q to be positive for airflow into the balloon, and used the fact that air moves from high to low pressure.

With the help of equation (2), Q can be expressed in terms of pressure differences:

Q=S2ρ(tLp0pLtHpHp0)
(4)

Using equation (1), pL can be traded in for pH to yield

Q=S2ρtH(pHp0)(tLtH)
(5)

Since the high-pressure phase is of shorter duration than the low-pressure phase, Q is positive; air flows into the kamifusen and it inflates.

I thank Yasushi Suzuki, Erik Ivins, and Bill Patzert for helpful suggestions.

1.
Logergist, “
Kamifusen no nazo wo toku
,” in
Shin butsuri no sanpomichi
, vol.
1
,
Chikuma Shobo
(
2009
), p.
210
.
2.
Logergist, “
Mimi ga itaku naru hanashi
,” in
Dai-san butsuri no sanpomichi
,
Iwanami Shoten
(
1966
), p.
119
.

Ichiro Fukumori is a principal scientist in the Earth science section at the Jet Propulsion Laboratory, California Institute of Technology, in Pasadena.