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 ID: 208038
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Slide1
Geometric Analysis of Shell MorphologySlide2
The universe is written in the language of mathematicsGalileo Galilei, 1623Quantitative analysis of natural phenomena is at the heart of scientific inquiryNature provides a tangible context for mathematics instruction
Math & NatureSlide3
ContextThe part of a text or statement that surrounds a particular word or passage and determines its meaning.
The
circumstances in which an event occurs; a setting
.
The Importance of ContextSlide4
Context-Specific Learning Facilitates experiential and associative learningDemonstration, activation, application, task-centered, and integration principles (Merrill 2002)
Facilitates generalization of principles to other contexts
The Importance of ContextSlide5
Geometry & BiologyBiological structures vary greatly in geometry and therefore represent a platform for geometric educationGeometric variability functional variability ecological variability
Mechanism for illustrating the consequences of geometry
Math & NatureSlide6
Morphospace is the range of possible geometries found in organismsVariability of ellipse axes
Math & NatureSlide7
Morphospace is the range
of possible geometries
found in organisms
Bird wing geometry
Math & NatureSlide8
Morphospace is the range of possible geometries found in organismsWhy are only certain portions of
morphospace
occupied?
Evolution has not produced all possible geometries
Extinction has eliminated certain geometriesFunctional constraints exist on morphology
Math & NatureSlide9
Morphospace is the range of possible geometries found in organismsFunctional constraints on morphology
Bird wing shape
Math & NatureSlide10
Morphospace is the range of possible geometries found in organismsAdaptive peaks in morphospace
“Life is a high-country adventure” (Kauffman 1995)
Math & Nature
McGhee 2006Slide11
Morphospace
is
the
range
of possible geometries
found in organisms
Adaptive peaks in
morphospace
Convergent evolution
Math & Nature
Reece et al. 2009Slide12
Spiral shell geometryConvergent evolutionMath & Nature
Cephalopods
Foraminiferans
Gastropods
Nautilids
soer.justice.tas.gov.au
,
www.nationalgeographic.com,
alfaenterprises.blogspot.comSlide13
Spiral shell geometryCephalopods past & presentMath & Nature
Nautilids
Ammonites
lgffoundation.cfsites.org,
www.nationalgeographic.comSlide14
Ammonite shell geometrySize Math & Nature
www.dailykos.comSlide15
Ammonite shell geometryShape
Math & Nature
lgffoundation.cfsites.org,
www.paleoart.com,
www.fossilrealm.comSlide16
Ammonite shell geometrySpiral dimensionsW = whorl expansion rate↓ W ↑ W
D = distance from axis
↓
D
↑ DT = translation rate
↓
T
↑
T
Math & NatureSlide17
Ammonite shell geometrySpiral dimensionsMath & Nature
Raup
1966
W
D
TSlide18
Ammonite shell geometryMorphospaceMath & Nature
McGhee 2006Slide19
Ammonite shell geometryWhich parts of ammonite morphospace are most occupied?
Math & Nature
Raup
1967Slide20
Ammonite shell geometryWhich parts of ammonite morphospace are most occupied?
The parts with overlapping whorls
Why?
Locomotion
Math & NatureSlide21
Ammonite shell geometryHow has ammonite morphospace changed over evolutionary history?Sea level changesShallow water forms
Deep water forms
Math & Nature
Bayer & McGhee 1984Slide22
Ammonite shell geometryHow has ammonite
morphospace
changed over evolutionary history?
Sea level changes
Shallow water formsDeep water forms
Convergent evolution x 3
Why?
↑ coiling = ↑ strength in deep (high pressure) environment
↓ ornamentation = ↓ drag in shallow (high flow) environment
Math & Nature
McGhee 2006Slide23
Ammonite shell geometryWhat happened when the ammonites went extinct?Nautilids invaded their morphospace!
Math & Nature
Ward 1980
XSlide24
Ammonite shell geometryQuestionHow do shell surface area and volume differ among ammonites with overlapping and non-overlapping whorls?
Math & Nature
lgffoundation.cfsites.org,
www.paleoart.comSlide25
Geometry & BiologyFlorida StandardsMAFS.912.G-GMD.1.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Math & NatureSlide26
Ammonite shell modelsNon-overlapping and overlapping whorlsProcedureUse clay to create two shell models of equal size
Measure their height and radius
Math & NatureSlide27
Ammonite shell modelsNon-overlapping and overlapping whorlsProcedureTwist one of the cones into a model with non-overlapping whorls and the other into a model with overlapping whorls
Math & NatureSlide28
Ammonite shell modelsNon-overlapping and overlapping whorlsProcedureMeasure the height and radius of the model with overlapping whorls
Math & NatureSlide29
Ammonite shell modelsNon-overlapping and overlapping whorlsProcedure
Calculate the volume of the space where the organism lives using the measurements of the original cones
Sample
data:
Math & NatureSlide30
Ammonite shell modelsNon-overlapping and overlapping whorlsProcedure
Calculate the surface area of both cones
Note that the surface area of the cone with non-overlapping whorls will be the same as the surface area of the original cone
Sample data:
Non-overlapping whorls
Overlapping whorls
Math & NatureSlide31
Ammonite shell modelsNon-overlapping and overlapping whorlsProcedure
Calculate the surface area-to-volume ratio for each shell model
Sample data:
Non-o
verlapping whorls
7
Overlapping whorls
Determine
which cone model has better swimming efficiency (i.e., less drag due to surface area
)
Math & NatureSlide32
Ammonite shell modelsAdditional workSurface area and volume comparisons of deep water and shallow water shell typesQuestionWhy is ornamentation less common in shallow (low flow) environment?
Math & Nature
www.fossilrealm.com,
www.thefossilstore.comSlide33
Ammonite shell modelsAdditional workSurface area and volume comparisons of deep water and shallow water shell typesProcedureSimulate ornamentation by adding geometric
objects to surface of cone and
measuring
changes in surface
area
Math & Nature
www.fossilrealm.comSlide34
ReferencesBayer, U. and McGhee, G.R. (1984). Iterative evolution of Middle Jurassic ammonite faunas. Lethaia. 17: 1-16.
Kauffman, S. (1995).
At Home in the Universe: The Search for Laws of Self-Organization and Complexity
. Oxford University Press.
McGhee, G.R. (2006).
The Geometry of Evolution
. Cambridge University Press.
Raup
, D. M.
(1966). Geometric
analysis of shell coiling:
general problems
.
Journal of Paleontology
. 40: 1178 – 1190.
Raup
, D. M.
(1967). Geometric
analysis of shell coiling:
coiling in
ammonoids
.
Journal of
Paleontology
. 41: 42-65
.
Reece
, J.B.,
Urry
, L.A., Cain, M.L., Wasserman, S.A.,
Minorsky
, P.V., and Jackson, R.B. (2009). Campbell Biology
, 9th
Edition. Benjamin Cummings. San Francisco, CA.
Math & Nature