The Optimal Can an uncanny approach *

1. The Optimal Canan uncanny approach* Carlos Zambrano & Michael Campbell *spiced up with concepts from statistical mechanics… This talk is dedicated to Dr. Gerald Gannon for inspiring research that is deep and accessible. v 2014.02.26.00

2. naively, least cost from minimal surface area • The variables that characterize the shape of a cylindrical can are radius and height • Constraint: to store a required amount of some product, the can must have a fixed volume • We want minimum Surface Area: • Minimize • Result: Can with • But cans in a store look more like this… what gives?

3. material cost formula • Body thickness of a can: (0.2mm)* • Cost of Tin Plate Steel to make can: • Density: * * • Cost: US $0.363 to $0.476 per pound * * * • Assumptions (for simplicity) – the cost of: • making a can is proportional to the volume of tin to make the can • tin plate steel is $0.40 per pound from * * *above • For a cylindrical can of radius and height , we’ll consider the ratio . • Then • - material cost per in2 of surface area to make can • Sheet of tin plate steel will be 0.008 in thick • Volume (in3) is then the surface area times 0.008 Bulk of sheets of tin plate steel used to make cans * www.alibaba.com/product-gs/656173961/metal_food_can.html ** www.emirapackaging.com/english/ASTM-A624M-84.htm ***www.alibaba.com/product-gs/555207963/tinplate_for_two_piece_can.html

4. storage costs“A simple rule to keep in mind is the larger the quantity you haveshipped at once, the less you will pay per containerfor freight.” * • Rectangular Storage – consider two situations • Rectangular cabinets: real estate / rental costs • 18-Wheeler Cargo Trailers: local delivery costs is a flat fee $150 • A few points about cans in rectangular storage • there will always be a necessary amount of unused volume because a can does not have a rectangular top and bottom (can opener requirement) • we can see below that the dimensions of the cargo trailer will result in some radii and heights being more efficient • we will only consider practical dimensions for cans (Table 1) *http://www.skolnik.com/faq.php

5. storage

6. storage possibilities (20’x8’x8’6”)

7. calculation of storage cost

8. maximum cans in storage space:the built-in discontinuity

9. total shipping cost per volume of canned goods • Suppose you need to deliver 300 cans of volume 2 • you have to pay $150, independently of the number of cans, to a truck driver to make a single delivery trip to a grocery store • Suppose the can size is efficient and you can fit all 300 cans in the truck • Then the shipping cost per volume of goods delivered is • per • For other can sizes with same volume, say you can only fit • halfthe number of cans in the truck, then • only half the volume of goods are delivered for a shipping cost (per unit volume) of • per • one-thirdthe original amount in the truck • shipping cost (per unit volume) would be • per • Shipping cost (note 2nd is per-can quantity) • where is the capacity-volume of a single can and is the max # of cans that will fit in the cargo trailer

10. the model:material + storage/shipping cost • material cost (per can) • cost of storage/shipping (per can) • where is the total cost for shipping or storing all cans • model - total cost to produce + store/ship can • stat mech model: internal energy + β·field energy

11. stat mech… phase transition? • Interesting case: retail manhattan, new york • lower Fifth Avenue between 42nd and 49th streets • $900 per square foot real estate cost (rent/lease)* • Average Retail Rent ($ per ft3 , assuming 8 ft ceilings) • E.g., cupboard that measures 4 ft in length (across), 1 foot in width (distance to back of cupboard), and 1 foot high has a volume of 4 • amount of money the tenant in a retail shop on Lower Fifth Avenue would be per month! • we can consider that to be equivalent to the Cost of Distribution($ per Delivery) *Spring 2012, c.f., “The Heart of Fifth Avenue Shopping is Heading to the South”, N.Y. Times, 4 Sep. 2012

12. computational analysis • we can use GEOGEBRA to examine a few optimal values of (those that minimize total cost) for given • You can enter numbers using the online app: http://www.geogebratube.org/student/m24814 • parameters for retail manhattan:

13. some final thoughts on phase transitions and symmetry breaking • a fascinating feature is the symmetry of the can (diam. = height) • low storage costs : material cost dominates optimal design • for higher storage costs , we see that the can is rectangular, and thus the discrete 90-degree rotational symmetry of the can (front profile) is broken • this is typical of a second-order phase transition in statistical mechanics • first-order phase transitions have a discontinuity • second-order transitions are continuous, • but the derivative is discontinuous • these are associated with symmetry-breaking • so we speculate that completing the graph of storage cost versus (numerically) will yield a continuous graph with a discontinuity in the derivative (likely a corner)… In other words, that this model displays a “second-order phase transition”

14. epilog: phase transitions & symmetry breaking • a phase transitionis the transformation of a system from one state (of matter, existence, etc) to another • water freezing into ice or vaporizing to steam • liquid crystals aligning in the nematic phase, so they will work in an LCD television • the early universe cooling to break the electroweak symmetry, resulting in the electromagnetic and weak forces decoupling into two “distinct” forces • phase transitions are characterized by a “non-analyticity” in a function called the “free energy” which measures the amount of work a (thermodynamic) system can perform • some examples of non-analytic indicators – a discontinuity in the: function, or its derivative, or its second derivative, …, or a function that’s infinitely differentiable with taylor series not matching the function (at one or more points)

15. mechanism of phase transitions • an order parameter is a function of a physical parameter describing a property of the system which typically is zero in one phase and non-zero in another • the magnetization of a ferromagnetic system is an order parameter (function of temperature)which is zero above a certain temperature (no magnetic properties), and non-zero below that temperature (magnetic properties) • if there is a (theoretical) phase transition, it is the result of a non-analyticity in the free energy • a visual cue for a phase transition frequently (but not always) is a break or corner/cusp on the graph of the order parameter as a function of the physical parameter

16. graph indicative of afirst-order phase transition • first-order phase transition: liquid crystals * • graph of light absorption versus temperature • note the discontinuity at 110 degrees C • as T increases, crystals change from a nematic(liquid-solid) state to an isotropic (liquid state) at the clearing point transition temperature (77 deg C here) which is also in fact discontinuous and first-order * Fig 2.8.1b, “Investigation of Parameters of Liquid Crystal Composite System”, 2nd ICWET 2011 Proc., Int. Jour. of Computer Appl.

17. graph indicative of asecond-order phase transition • second-order phase transition: 2D Isingferromagnet* • graph of Spontaneous Magnetization (M) versus Temperature (T) • note M is a continuous function of T and approaches Tc≈ 2.27 with infinite slope • at Tc , the derivativeχ(susceptibilty) diverges andisdiscontinuous • alignment of spins change from aligned (magnetic) state to a paramagnetic (nonmagnetic) state at the critical temperature Tc • this is continuous and 2nd-order behavior • ferromagnetic phase • low temperature • broken symmetry – flipping all spins yields different result (spins down) • paramagnetic phase • high temperature • symmetry – flipping all spins yields same (random directions) * simulation at http://www.ibiblio.org/e-notes/Perc/trans.htm

18. phase order of optimal can • since a symmetry is broken for the optimal can (at high storage/shipping costs), we suspect a continuous transition (non-first-order: 2ndor 3rd or…) transition and speculate • looking at the order parameter (value of that minimizes total cost of the can) as a function of (flat storage cost rate) • there is a critical cost • below which : the can has the symmetry of a square front profile • above which : the symmetry is broken and the front profile is rectangular like the ones we see in grocery stores • ) is continuous, but non-analytic at ; i.e., the optimal can model has a continuous phase transition

19. computational analysis of optimal can model • we can use the GEOGEBRA program to plot values of versus to find the critical value via a possible corner/kink (or discontinuity if 1st order) in the graph • see the previous slide and test various “Cost Distribution” values (i.e., values) • see what the resulting values for (by observing when doptsignificantly differs from hopt ) • we are interested in the minimum on the red curve (total cost); the blue curve is the material cost only (the standard one done in calculus)

20. procedure to plot versus • go to http://www.geogebratube.org/student/m24814 • Enter the following parameters, followed by return key: • enter diam=3.7821, height=3.769, for example to set the volume to that of a standard No. 2 can ( ) • set the volume of the storage cabinet: length=36, width=12, height=12 • the cost of tin should be 0.0009076 • enter Cost to Ship =$ 60 • Note the values for doptand hopt are returned and • Here’s where it gets fascinating: • doptand hopt stay roughly equal up to $66.5978 and we get for with critical storage cost • doptand hoptstart to differ for > $66.5978 and we get a large jump for • The symmetry is broken above a critical cost and this reeks of phase transition behavior!

21. open problem • of course now the problem is to create a very fine set of values for around the critical cost • what does the graph look like? • we speculate it is continuous with a kink or cusp at • i.e., there is a continuous phase transition because the front-profile square symmetry of the can is broken above • Finally, there may be an analytical way to examine the minimum using a “free-energy”-type approach via the saddle-point method. But this is beyond the scope of this presentation. • Let me know what you find: aller au fond des choses!