1
Length of any side = 1L
Area of any face = A
A = L
2
Volume = V
V = L
3
A = V
2/3
= L
2
Length of any side = 3L
Area of any face = A = 9L
2
Volume = V = 27L
3
A = V
2/3
= 9L
2
In general: A
∝
L
2
, V
, V
∝
L
3
,
and
and
A
A
∝
V
2/3
Allometry is nothing more than a detailed
analysis of the effects of scale, or size.
L
A = L
2
3L
A = 9L
2
Allometry is nothing more than a detailed
analysis of the effects of scale, or size.
Allometric relationships are expressed as power functions,
where a variable of interest, such as metabolic rate, is
expressed as a power function of body size.
Power functions are of the form: Y = aX
b
, where X is
raised to the power b.
Power functions are nonlinear and can be hard to visualize
and manipulate.
Log transformation is used to linearize power functions.
From: Y = aX
b
To: log Y = log a + b(log x).
The logtransformed equation describes a straight line.
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2
y = x
2
R
2
= 1
0
50
100
150
200
250
300
350
400
0
5
10
15
20
Length
Area
y = x
3
R
2
= 1
0
1000
2000
3000
4000
5000
6000
7000
8000
0
5
10
15
20
Length
Volume
0
1000
2000
3000
4000
5000
6000
7000
8000
0
5
10
15
20
Length
Area or Volume
y = 2x + 0
R
2
= 1
0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Log Length
Log Area
y = 3x + 0
R
2
= 1
1
1.5
2
2.5
3
3.5
4
4.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Log Length
Log Volume
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Log Length
Log Area or Log Volume
y =
aX
b
log y = log a + b log X
log transformation
J
equation of a straight line
Area = a
●
(Length)
2
Log (Area) = Log (a) + 2
●
Log (Length)
Volume = a
●
(Length)
3
Log (Volume) = Log (a) + 3
●
Log (Length)
2
2
3
2
0
1000
2000
3000
4000
5000
6000
7000
8000
0
5
10
15
20
Length
Area or Volume
2
2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Log Length
Log Area or Log Volume
y =
y =
aX
aX
b
log y = log a + b log X
log y = log a + b log X
log transformation
J
equation of a straight line
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 Spring '11
 Staff
 Ecology, metabolic rate, Warmblooded, individual body mass

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