Geometric Analysis of Suction Feeding. The universe is written in the language of mathematics ◦...

Geometric Analysis of Suction Feeding

The universe is written in the language of mathematics◦ Galileo Galilei, 1623

Quantitative analysis of natural phenomena is at the heart of scientific inquiry

Nature provides a tangible context for mathematics instruction

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Context

1. The part of a text or statement that surrounds a particular word or passage and determines its meaning.

2. The circumstances in which an event occurs; a setting.

The Importance of Context

Context-Specific Learning

◦ Facilitates experiential and associative learning

Demonstration, activation, application, task-centered, and integration principles (Merrill 2002)

◦ Facilitates generalization of principles to other contexts

The Importance of Context

Geometry & Biology◦ Biological structures vary greatly in geometry and

therefore represent a platform for geometric education

◦ Geometric variability functional variability ecological variability Mechanism for illustrating the consequences of

geometry

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Vertebrate skulls vary greatly in form & function

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www.digimorph.org

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csi.whoi.edu www.digimorph.org

22 bones1 moving part

~50 bones~7 moving parts

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Liem et al. (2001)

Vertebrate skulls vary greatly in form & function◦ Moveable parts of the fish skull are responsible for

the diversity of feeding mechanisms in fish Jaw protrusion in the sand tiger shark Carcharias

taurus

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D. Huber

Vertebrate skulls vary greatly in form & function◦ Moveable parts of the fish skull are responsible for

the diversity of feeding mechanisms in fish Jaw protrusion in the sling-jaw wrasse Epibulus

insidiator

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P. Wainwright

Fish feeding mechanisms◦ Filter ◦ Biting◦ Ram ◦ Suction

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www.true-wildlife.blogspot.com

C. Fallows

www.z00n.net

Fish feeding mechanisms◦ Filter feeding

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W. Mischler 2013

Motta et al. (2010)

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Motta et al. (2010)

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P. Motta

Fish feeding mechanisms◦ Biting

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Fish feeding mechanisms◦ Biting

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www.digimorph.org

Fish feeding mechanisms◦ Ram feeding

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D. Huber C. Fallows

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S. Huskey www.tennesseeaquarium.com

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D. Huber

Fish feeding mechanisms◦ Suction feeding

Most common fish feeding mechanism Water cohesion Suction performance

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D. Huber

http://www.youtube.com/user/Wainwrightlab

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Wainwright et al (2006)

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Svanback et al (2002)

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Grubich (2001)

Anterior posterior expansion

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Gibb & Ferry-Graham (2005)Wainwright et al (2006)

Fluid flow

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Holzman et al (2008)

Fluid pressure and movement speed

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Svanback et al (2002)

Feeding ecology

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Motta et al (2008)

Geometric modeling

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Van Wassenbergh et al (2007)Bishop et al (2008)

Goliath grouper Epinephelus itajara

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Goliath grouper Epinephelus itajara Questions

What fluid velocity can the goliath grouper generate during suction feeding?

How does suction feeding by the goliath grouper compare to other fish?

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Geometry & Biology◦ NGSSS

MA.912.G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and area.

MA.912.G.5.4: Solve real-world problems involving right triangles.

MA.912.G.7.5: Explain and use formulas for lateral area, surface area, and volume of solids.

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Geometry & Biology◦ NGSSS

MA.912.G.7.7: Determine how changes in dimension affect the surface area and volume of common geometric solids.

MA.912.G.8.2: Use a variety of problem solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards.

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Geometry & Biology◦ CCSS

MACC.912.G-GMD.1.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

MACC.912.G-GMD.2.4: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

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Geometry & Biology◦ CCSS

MACC.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

MACC.K12.MP.1.1: Make sense of problems and persevere in solving them.

MACC.K12.MP.4.1: Model with mathematics

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Goliath grouper model◦ Objective

Determine the velocity of water flow into the mouth

◦ Procedure Determine the volume of components A and B at rest

(t0) and at maximum expansion (t1) t0 = time at rest

t1 = time at maximum expansion

Determine the volume change during feeding

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◦ Procedure Determine the area of the mouth at maximum

expansion (t1) t1 = time at maximum expansion

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◦ Procedure

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Suction feeding in the goliath grouper◦ Given

Dimensions of cones A and B at rest (t0)

1) Find the volume of the goliath grouper feeding mechanism at rest (t0).

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Dimensions of cones A and B at rest (t0)

1) Find the volume of the goliath grouper feeding mechanism at rest (t0).

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Goliath Grouper Suction Feeding

Time 0

Cone A Length (mm) Area (mm2) Volume (mm3)a 34.9

N/A b 153.6

Cone B Length (mm) Area (mm2) Volume (mm3)a 34.9

N/A c 54.3d 6.4e

Volume of feeding mechanism before expansion (t0)

Time 1

Cone A Length (mm) Area (mm2) Volume (mm3)a

Cone B Length (mm) Area (mm2) Volume (mm3)a

Volume of feeding mechanism at maximum expansion (t1)

Volume change during feeding event (mm3) Duration of feeding event (sec) 0.132

Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec)

Time 0

N/A 195916.8b 153.6

N/A 84289.7c 54.3d 6.4e 12.2

Volume of feeding mechanism before expansion (t0) 180206.5

Time 1

Cone A Length (mm) Area (mm2) Volume (mm3)a

Cone B Length (mm) Area (mm2) Volume (mm3)a

Dimensions of cones A and B at maximum expansion (t1)

2) Find the volume of the goliath grouper feeding mechanism at maximum expansion (t1).

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Time 0

N/A 195916.8 b 153.6

N/A 84289.7 c 54.3d 6.4e 12.2

Time 1

N/A b 161.3

c 56.4d 32.6e

Time 0

N/A 195916.8 b 153.6

N/A 84289.7 c 54.3d 6.4e 12.2

Time 1

N/A 263547.1b 161.3

230974.7c 56.4d 32.6e 266.5

494521.7

Volume change during feeding event (mm3) 214315.3Duration of feeding event (sec) 0.132

Dimensions of cone B at maximum expansion (t1)

3) Find the area of the goliath grouper mouth at maximum expansion (t1).

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A. Collins

Time 0

N/A 195916.8 b 153.6

N/A 84289.7 c 54.3d 6.4e 12.2

Time 1

N/A 263547.1b 161.3

230974.7c 56.4d 32.6e 266.5

494521.7

Time 0

N/A 195916.8 b 153.6

N/A 84289.7 c 54.3d 6.4e 12.2

Time 1

N/A 263547.1b 161.3

3338.8 230974.7c 56.4d 32.6e 266.5

494521.7

Area of mouth at maximum expansion (t1) (mm2) 3338.8 Velocity of water flow into mouth (mm/sec)

Volume of the goliath grouper feeding mechanism at rest (t0) and at maximum expansion (t1)

Duration of the feeding event (t1 - t0)

Area of the mouth opening at maximum expansion (t1)

4) Find the velocity of water flow into the mouth of the goliath grouper during suction feeding.

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Time 0

N/A 195916.8 b 153.6

N/A 84289.7 c 54.3d 6.4e 12.2

Time 1

N/A 263547.1b 161.3

3338.8 230974.7c 56.4d 32.6e 266.5

494521.7

Area of mouth at maximum expansion (t1) (mm2) 3338.8 Velocity of water flow into mouth (mm/sec) 486

Wainwright et al (2006)

Suction feeding in the snook Centropomus undecimalis

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Korhnak JJ Photo

Suction feeding in the snook Centropomus undecimalis◦ Given

Dimensions of cones A and B at rest (t0) and at maximum expansion of the feeding mechanism (t1)

5) Find the velocity of water flow into the mouth of the snook during suction feeding.

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Snook Suction Feeding

Time 0

N/A b 27.6

N/A c 12.3d 1.8e

Time 1

N/A b 28.9

c 12.3d 5.9e

Snook Suction Feeding

Time 0

N/A 127.5 b 27.6

N/A 147.2c 12.3d 1.8e 73.8

Time 1

N/A 1482.9 b 28.9

109.4 1611.5c 12.3d 5.9e 66.0

3094.4

Area of mouth at maximum expansion (t1) (mm2) 109.4Velocity of water flow into mouth (mm/sec) 716

Suction feeding in the longjaw butterfly fish Forcipiger longirostris

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Ferry-Graham et al (2001)

S. Huskey

Suction feeding in the longjaw butterfly fish Forcipiger longirostris◦ Given

Dimensions of cones A and B at rest (t0) and at maximum expansion of the feeding mechanism (t1)

6) Find the velocity of water flow into the mouth of the longjaw butterfly fish during suction feeding.

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Longjaw Butterfly Fish Suction Feeding

Time 0

N/A b 14.9

N/A c 31.2d 1.1e

Time 1

N/A b 14.9

c 31.6d 1.1e

Longjaw Butterfly Fish Suction Feeding

Time 0

N/A 390.1 b 14.9

N/A 1036.0c 31.2d 1.1e 8.8

Time 1

N/A 390.1b 14.9

3.8 1049.3c 31.6d 1.1e 8.9

1439.4

Area of mouth at maximum expansion (t1) (mm2) 3.8Velocity of water flow into mouth (mm/sec) 159

Suction feeding ◦ Given

Velocities of water flow into the mouths of all three fish

7) Determine which fish is the best suction feeder.

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Korhnak

v(t) = 159 mm/secv(t) = 716 mm/secv(t) = 486 mm/sec

References◦ Bishop, K.L., Wainwright, P.C., and Holzman, R. (2008). Anterior to posterior wave of buccal

expansion in suction feeding fish is critical for optimizing fluid flow velocity profile. Journal of the Royal Society, Interface. 5:1309-1316.

◦ Ferry-Graham, L.A., Wainwright, P.C., and Bellwood, D.R. (2001). Prey capture in long-jawed butterflyfishes (Chaetodontidae): the functional basis of novel feeding habits. Journal of Experimental Marine Biology and Ecology. 256:167-184.

◦ Galileo Galilei, The Assayer, as translated by Stillman Drake (1957), Discoveries and Opinions of Galileo pp. 237 - 238. New York: Doubleday & Company.

◦ Gibb, A.C. and Ferry-Graham, L.A. (2005). Cranial movements during suction feeding in teleost fishes: Are they modified to enhance suction production? Zoology. 108(2): 141-153.

◦ Grubich, J.R. (2001). Prey Capture in Actinopterygian Fishes: A Review of Suction Feeding Motor Patterns with New Evidence from an Elopomorph Fish, Megalops atlanticus. Integrative and Comparative Biology. 41(6): 1258-1265.

◦ Holzman, R., Day, S.W., and Wainwright, P.C. (2007). Timing is everything: coordination of strike kinematics affects the force exerted by suction feeding fish on attached prey. Journal of Experimental Biology. 210: 3328-3336.

◦ Holzman, R., Day, S.W., Mehta, R.S., and Wainwright, P.C. (2008). Jaw protrusion enhances forces exerted on prey by suction feeding fishes. Journal of the Royal Society, Interface. 5(29): 1445-1457.

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References◦ Liem, K., Bemis, W., Walker, W.F., and Grande, L. (2001). Functional Anatomy of the

Vertebrates: An Evolutionary Perspective. New York. Cengage Learning. ◦ Merrill, M.D. (2002). First principles of instruction. Educational Technology Research and

Development. 50 (3): 43 – 59.◦ Motta, P.J., Hueter, R.E., Tricas, T.C., Summers, A.P., Huber, D.R., Lowry, D., Mara, K.R.,

Matott, M.P., Whitenack, L.B., and Wintzer, A.P. (2008). Functional morphology of the feeding apparatus, feeding constraints, and suction performance in the nurse shark Ginglymostoma cirratum. Journal of Morphology. 269(9): 1041-1055.

◦ Motta, P.J., Maslanka, M., Hueter, R.E., Davis, R.L., de la Parra, R., Mulvany, S.L., Habegger, M.L., Strother, J.A., Mara, K.R., Gardiner, J.M., Tyminski, J.P., and Zeigler, L.D. (2010). Feeding anatomy, filter-feeding rate, and diet of whale sharks Rhincodon typus during surface ram filter feeding off the Yucatan Peninsula, Mexico. Zoology. 113: 199-212.

◦ Sanford, C.P.J. and Wainwright, P.C. (2002). Use of sonomicrometry demonstrates the link between prey capture kinematics and suction pressure in largemouth bass. Journal of Experimental Biology. 205: 3445-3457.

◦ Svanback, R., Wainwright, P.C., and Ferry-Graham, L.A. (2002). Linking cranial kinematics, buccal pressure, and suction feeding performance in largemouth bass. Physiological and Biochemical Zoology. 75(6): 532-543.

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References◦ Van Wassenbergh, S., Herrel, A., Adriaens, D., and Aerts, P. (2007). No trade-off between

biting and suction feeding performance in clariid catfishes. Journal of Experimental Biology. 210: 27-36.

◦ Wainwright, P.C., Huskey, S.H., Turingan, R.G., and Carroll, A.M. (2006). Ontogeny of suction feeding capacity in snook, Centropomis undecimalis. Journal of Experimental Zoology. 305A: 246-252.

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Calculating the volume of a truncated cone

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Geometric Analysis of Suction Feeding. The universe is written in the language of mathematics ◦...

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Transcript of Geometric Analysis of Suction Feeding. The universe is written in the language of mathematics ◦...

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