Patterns in Gastropod Shape and
Form: Snails Can Do the Math!
Alice J. Monroe
St. Petersburg College and
Suncoast Conchologists
Clearwater, Florida
Presentation Overview
• Snail anatomy
– a short biological introduction
• How a snail makes a shell
• Seashells:
– variations on a spiral theme
• Colors and patterns in seashells
• Significance and applications of
understanding seashell form
Body Plan of a Snail
heart
mantle cavity
anus
gill
mantle
digestive
gland
No brain
foot
radula
The mantle
makes the shell.
Mantle
Mantle Shell
The mantle cells secrete
calcium carbonate to build
the shell.
Seashells are Variations
on a Spiral Theme
A Spiral Can Be Described
Mathematically
• Begin with a circle
• Drag it through space
• Enlarge the circle as it is dragged
• Coil it around a central axis
3 Parameters Describe a Shell
• W=Width. Describes the rate at which the diameter of
the tube grows. Higher values mean the opening of the
shell becomes wider with each rotation.
• D=Distance. The distance of the center of the tube from
the axis of coiling. Higher values mean the tube of the
shell forms farther from the coiling axis.
• T=Tallness. The distance of the center of the tube from
the previous rotation along the axis of rotation. Higher
values mean the shell is taller. The effect of T depends
on the change in this width, so if there is no change
(W=0), then the shell doesn't coil at all but forms a torus
(a doughnut)!
A shell-generating applet demonstrates how these parameters
affect the growing shell.
X-rays of shells reveal the internal
spiral shape.
Shells are made of calcium
carbonate, similar to bone. They are
wonderful candidates for x-rays.
Spirals can be described mathematically in 2 dimensions
Many names:
Logarithmic spiral
Equiangular spiral
Geometric spiral
Growth spiral
Spira mirabilis
Golden spiral is a
special case
Golden spiral is a
Fibonacci series.
1 + 1 = 2; 1 + 2 = 3; 2 + 3 = 5; 3 + 5 = 8; 5 + 8 = 13
An Equiangular Spiral
Can Snails Do Math?
• Snails do not think in terms of
mathematical equations—no brains!
• Snails build shells that enable them to
survive in their environments
• Biologists impose math on snails
• Mathematical descriptions of shell shape
indicate that can be predictable
• Shell shape can be modeled
Computer-modeled shells show strong
resemblance to real shells.
But what about worm
shells and other
gastropods that seem not
to know the math?
What about colors?
Pigments are
incorporated into
the shell as it is
being formed.
The mantle
makes the shell.
Pigments are
incorporated at
the same time.
Mantle
Mantle Shell
The mantle cells
secrete calcium
carbonate to build the
shell.
The mantle cells
secrete pigment
molecules at the same
time.
Calcium carbonate is
white. Pigments give
shells other colors.
What are Pigments?
• Protein molecules that are byproducts of
metabolism, ie. waste
• They originate from plants that the animals
eat
– Example: carrots have b-carotene, an orange
pigment
• Snails incorporate these waste materials
into their shells—(recycle-reuse)
How Do Pigments Get Into the
Shells?
• Shells are made of calcium carbonate
arranged in layers like bricks
• Pigments are secreted along with the
calcium carbonate
• Pigments are incorporated into the shells
like mortar in between the bricks
Why Pigments in Shells?
• Colors are often not conspicuous
– Many shells live in deep water—colors are not
visible at depths with no light penetration
– Many shells have thick external coverings, or
encrusting organisms
• Possible explanations:
– Anti-pollution strategy
– Cryptic coloration for predator avoidance
– Species preservation in a crowd—the jelly
bean hypothesis
Pigments secreted to yield solid colored shells.
Mantle cells that secrete pigments
Growing shell edge
Growing shell edge
Growing shell edge
All cells secreting pigment all
the time results in a solid
color shell.
Most shells are not solid colors; most shells have
patterns. Patterns can be mathematically described.
The formation of spiral line patterns on shells.
Mantle cells that secrete pigments
Growing shell edge
Growing shell edge
Discontinuous spiral line patterns form dots or dashes.
The formation of axial line patterns on shells.
Mantle cells that secrete pigments
Growing shell edge
Growing shell edge
Mathematical descriptions of patterns
enable computer-generated models of
patterns and shells.
The Tent Pattern: A Fractal
Whether found in nature or created in art,
fractals consist of a rough or fragmented
geometric shape that can be subdivided
into parts, each of which is (at least
approximately) a reduced-size copy of the
whole. Fractals are generally self-similar
and independent of scale.
Pigments secreted to yield the tent pattern.
Mantle cells that secrete pigments
Growing shell edge
Pigments secreted to yield the tent pattern.
Mantle cells that secrete pigments
Growing shell edge
Pigments secreted to yield the tent pattern.
Mantle cells that secrete pigments
Growing shell edge
Pigments secreted to yield the tent pattern.
Mantle cells that secrete pigments
Growing shell edge
Pigments secreted to yield the tent pattern.
Mantle cells that secrete pigments
Growing shell edge
Pigments secreted to yield the tent pattern.
Mantle cells that secrete pigments
Growing shell edge
Pigments secreted to yield the tent pattern.
Mantle cells that secrete pigments
Growing shell edge
Pigments secreted to yield the tent pattern.
Mantle cells that secrete pigments
Growing shell edge
Variations on the Tent Pattern Theme
Ammonite Sutures: Another
Example of a Natural Fractal
“Real” Shells are not Perfect
• These are mathematical models of shell
growth and pattern formation
• Snails build shells while they are crawling
around
– Finding food
– Avoiding being someone else’s food
– Reproducing
• Conditions are not always ideal, so shells
are not “perfect” as the models might
suggest
Photographs on the Left;
Rendered Images on the Right
Seashell Form and Function
• To a biologist, seashell modeling provides
useful information about the animals, both
present and past
• Beyond biology, seashell modeling has
applications in other areas such as
architecture
Inspired by Seashells
Guggenheim Museum
Inspired by Seashells
Sydney Opera House
Inspired by Seashells
Van Wezel Performing Arts Hall
Pagoda
Summary and Conclusion
• Seashells are natural objects made by
snails
• Seashells have shapes and colors that
exhibit unity yet diversity
• Mathematics provides the tools to
understand the complexity of shape and
color patterns in seashells
• Seashells provide us with ideas to apply in
other areas such as architecture
So, What About the Math?
A Scientist Cannot Have Too Much
Mathematics
• Mathematics develops a thinking process that is
useful in many subject areas
• Taking mathematics courses is a luxury—
someone else explains it to you
• Recommended to me in Biology:
– Calculus with Analytical Geometry—3 courses
• For comprehension of 3-dimensional space
– Differential Equations
• Continuation of concepts of calculus
– Linear Algebra
• Matrix manipulation, specifically for Ecology
– Statistics—7 courses
• Experimental design and data analysis
The Algorithmic Beauty of
Seashells: Patterns in Growth and
Color
Whether found in nature or created in art, "fractals are a rough or
fragmented geometric shape that can be subdivided into parts, each of
which is (at least approximately) a reduced-size copy of the whole.
Fractals are generally self-similar and independent of scale."
The Algorithmic Beauty of Seashells:
Patterns in Growth and Color
Alice J. Monroe
Biology Department
St. Petersburg College
Biology as a Profession
• As a scientist, a biologist studies living
organisms in order to understand form and
function
• A biologist typically specializes in a
specific
– Group of organisms, example: Sharks
– Process, example: Nerve function
• A biologist seeks to explain a lot with
simplicity, yet elegance
• Mathematics provides tools to do this