While m&m's and Klein bottle geometry ostensibly have little technical or economic value in common, research was undertaken to establish theoretical and empirical values for the number of m&m's that could be contained in an Erlenmeyer Klein Flask (EKF). This research was performed solely for the purpose of obtaining such a flask absolutely free of charge, either with or without m&m's.
Four researchers at major midwest university
independently developed methodology to estimate the number of
m&m's contained in the aforementioned vessel, using data provide
by the original problem proposer. Three theoretical models were
developed to estimate values, using a standard Erlenmeyer flask as a
model for the Klein analog. An empirical measurement was used to
confirm the validity of all models, along with a statistical analysis
of m&m size.
Empirical measurement found that the EKF contains 549 +/- 3 m&m's. One theoretical model produced values within 1% of this number, but other ranged as high as 14% difference. However, there may be complicating factors in the method producing the higher error value.
Klein bottles are 2 dimensional topological structures that exist in 3 dimensions, having zero volume. However, they have been used for a variety of entertaining purposes, as noted in (1). Given the highly multi-disciplinary nature of this problem, it was thought that a single approach would not be as successful in developing useful solution. Therefore, we attempted to develop solutions using physical chemical, mathematical, sociological, and engineering approaches. The objective of this work was to obtain a completely free Erlenmeyer Klein Flask as noted in (4). However, if available, a completely free Klein Stein of similar volume (5) would be preferable.
Materials and methods
The Erlenmeyer flask used to approximate the Klein analog was obtained from Fischer Scientific (Pittsburgh, PA), model 4980, 500 mL Pyrex. Two packages of m&m's were obtained from a local grocery store (m&m/Mars, Hackettstown, NJ, net wt. 14.0 oz, milk chocolate). All calculations were performed on a Sharp Scientific calculator (model EL-5100S (Sharp Corp., Korea) using pre-installed algorithms for transcendental functions. 15 cm ruler used to measure m&m oblate spheroid radii was from Davis Liquid Crystals, Inc. (San Leandro, CA). All other materials used were of reagent grade or better.
Method 1: Physical Chemistry
Assume m&m's are oblate spheroids with a major axis radius of 0.6 cm (a) and a minor axis radius of 0.3 cm (b). The volume of oblate spheroid is (4/3)p a2b]. Assuming hexagonal close packing (hcp) packing, the void volume % of the packaging is (4/3)p a2b/(p/1.2092) a2(4)b = 0.4031 or 40.31% he available volume is composed of m&ms (2, 3).
Assuming the Erlenmeyer Klein bottle is basically a right angle cone, we can use its dimensions of 240 mm x 100 mm (hxD). The volume of a cone [(p/3)*(D/2)2*h] with these dimensions is 628.32 cm3.
This means the effective volume filled with m&m mass must be approximately 628.32*0.4031 = 253.26 cm3. Dividing this value by the volume of a single m&m (0.4524 cm3), gives the number of m&m in the bottle as 559.8.
Method 2: Mathematical estimation:
Alternatively, consider the specified volume of the flask. Given that the original volume of the Erlenmeyer flask is 500 cm3, this must be corrected to account for the volume of the neck, which is approximately 110 ml, so the total volume would be 610 cm3.
Multiplying this value by the void volume of hcp packed oblate spheroids (0.4031) gives a volume of 245.89 cm3 of m&m's. Dividing this volume by the the volume of a single m&m (0.45239 cm3) gives the total number of m&m's as 543.53.
Method 3: Liberal arts estimation
Fact: M&m's are mainly composed of chocolate.
Fact: Chocolate is recognized to have a transcendental power on bipedal mammals, particularly female humans.
Fact: The best known transcendental numbers are e and p.
Fact: Normal bipedal human female mammals have 20 digits, which are used to eat m&m's.
Fact: Erlenmeyer Klein Flask has 20 letters.
Armed with this knowledge, we can estimate the number of m&m's in the flask by multiplying the the number of bipedal mammalian digits used to ingest m&m's by the transcendental number e to the power of p and add number of letters in Erlenmeyer Klein Flask to obtain:
20*ep + 20 = 482.81
This demonstrates that without an extensive knowledge of mathematics or physical chemistry, one can also simply estimate the number of m&m's in an Erlenmeyer Klein flask. However, as normal, this also demonstrates that a liberal arts degree is essentially worthless in technical computations.
Method 4: Engineering estimation
2 Bags of m&m's were purchased and poured slowly into a 500 mL Erlenmeyer flask from the lab. This was followed by equilibrating the system using gentle agitation and addition of m&m's to fill to the lip of the vessel. Subsequently, the m&m's were poured out into large plastic weighing dishes and manually counted. Ignoring the broken and chipped ones, which were used for personal metabolic studies, the number obtained was 549 +/- 3.
Results and Discussion
Results of the 4 methods are summarized in Table 1.
Table 1. Raw data
Method 1, Chemist: 559.8
Method 2, Mathematician: 543.5
Method 3, Liberal Arts: 482.8
Method 4, Engineer: 549 +/- 3
There is inherent error in any of these
calculations or estimates, given that the angles and details of
elongation and extension of the neck of the flask to form the Klein
nexus are unknown (see Fig. 1). Unfortunately, accounting for these
errors is not possible without additional data on the geometrical
Klein Bottle nexus
It was found that although the assumption of an oblate spheroid for a single m&m is quite reasonable. Within the precision of the available instrumentation (my eye and a 15 cm ruler marked in 1 mm increments), the radii of the long and short axes of an m&m were precisely 0.6 cm and 0.3 cm, respectively. However, extensive statistical analysis of the contents of two 14.0 oz bags of m&m's, approximately 1100 pieces (aren't graduate students terrific!), demonstrated that the mean mass of m&m's is 0.911725 gm, with a standard error of 0.03174 gm. Ignoring the density changes between the candy coating vs. the chocolate interior, this converts to a potential error of 3.48% in the volumetric calculation of a single m&m. Using this variation, the numbers previously obtained by methods 1 and 2 must be adjusted to yield the following corrected values (see Table 2):
Table 2. Corrected values
Method 1, Chemist: 559.8 +/- 19.5
Method 2, Mathematician: 543.5 +/- 18.9
Method 3, Liberal Arts: 482.8
Method 4, Engineer: 549 +/- 3
There are several obvious conclusions that can be
developed from these results. First, the engineer's method (method 4)
is highly accurate under the current situation. It employs the fewest
number of assumptions. However, the method employed is wholly
empirical and does not account for the theoretical nature of m&m
structure, the complexities of m&m packing within the vessel, or
the statistical issues involving population variation among the
sample. Therefore, while valuable for the purposes of this contest,
it cannot be extrapolated to other shapes, sizes, or situations.
The chemist's method (method 1) clearly has the highest absolute error value, nearly 20 m&m's. Additionally, the assumption of hexagonal close packing (hcp) is clearly in error, given the graphical evidence shown on the image (see Fig. 1). The m&m's in the nexus clearly demonstrate body centered cubic packing (bcc), not hcp.
Method 3 yields results with a similar statistical error value, but gives a value that is approximately 3.0% lower than method 1. However, given the statistical error value, it is clearly close to the actual value as found in method 3. This is probably due to the greater accuracy involved with measuring the additional volume of the flask neck, vs. the assumption of a conical shape, as in method 1. Further improvement of method 1 might involve using a truncated cone assumption combined with a short right cylinder analysis to obtain an improved estimate of the additional volume of the neck of the flask.
Method 3 clearly has significant shortcomings vs. the other methods. This is not surprising, given the highly heuristic nature of the methodology employed. The value obtained is nearly 14% different from the values obtained by other methods, although the methodology employed is highly appealing and very simple. It does not account for structural geometry, physical chemistry, or statistical variation, and has very little sound theoretical mathematical basis. Additionally, the veracity of the experimentalist may be in question. This question was raised due to the discovery that following the experimental procedures, approximately 25% of the original mass of m&m's provided to the researcher were absent. She was noted in her lab book that this may have involved "cold fusion, high m&m vapor pressures, or other unspecified errors". Additionally, a lab assistant noted a mysterious brown smudge on the researcher's lips, although this information is purely anecdotal.
Further research in this area may involve extension of the theoretical models developed herein to other geometries of Klein bottles, notably ellipsoidal, cylinderical, and modified spiral, provided suitable research funding or free vessels could be obtained.
1. Anon., Acme Klein Bottle, http://www.kleinbottle.com/, Oakland, CA.
2. Castellan, G. W., Structures of Solids and Liquids, Chapt. 26, in Physical Chemistry, 2nd ed., Addison-Wesley, 1971, pp. 633-637.
3. Hoerl, A. E., Plane Geometric Figures with Straight Boundaries, Perry's Chemical Engineers' Handbook, 6th ed., McGraw Hill, 1984, p. 2-11,
4. http://www.kleinbottle.com/m%26ms_in_a_klein_bottle.htm, Acme Klein Bottle, Oakland, CA.
5. http://www.kleinbottle.com/drinking_mug_klein_bottle.htm, Acme Klein Bottle, Oakland, CA.