# 02_Allometry - Allometry page 1.01 RR Lew isometric example...

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Allometry – page 1.01 – RR Lew The word Allometry has two roots. Allo means ‘other’ and metric refers to measure- ment. The Oxford American Dictionary defines Allometry as “the growth of body parts at different rates, resulting in a change of body proportions.”. It is different from either isometry (where iso means ‘same’) or anisometry (where aniso means ‘not the same’). The emphasis is on a difference in shape or proportions. Allometry may be a term unique to biology, since it is biological organisms that must change their shape (and thus their relative proportions) when their size is changed. To give an example of the differences (and similarities) of allometry and isometry, here is a relatively simple example of isometry that gives insight into why biological organisms tend to be allo- metric. A cube has a surface area of 6 • L 2 . Its volume is L 3 . As long as the shape is constant, the ratio of suraface area to volume will always be (6 • L 2 ) / L 3 , or 6/L. For a sphere, the surface area is 4 • ± • r 2 , and the volume is ± • r 3 ; the corresponding ratio of surface area to volume is 4/r. L 2•r area volume ratio (6/L) area and volume ratio x • L The ratio of area to volume scales as the inverse of L with an increase in size. The graph (left) shows the area, volume and ratio as a function of the size of the cube (as a multiple of L). N ota bene: The decrease in the surface area to volume ratio has an extraordinary impact on biological organisms. Larger organisms have less surface area available for taking up nutrients and radiating heat generated in respiration. That is, both nutrient requirements and heat genera- tion scale with organismal volume, not area. But, nutrient uptake and heat loss scale with area. isometric example 0 60000 120000 180000 240000 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1

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Allometry – page 1.02 – RR Lew Besides nutrient uptake / utilization and metabolic processes / heat loss, another aspect of the consequences of isometry are related to the strength of materials. Re-casting the area / volume of a cube in a different way [1] : (area) A 1 = 6 • L 2 A k = 6 • (k • L) 2 A k = 6 • k 2 • L 2 ( = k 2 • A 1 ) (volume) V 1 = L 3 V k = (k • L) 3 V k = k 3 • L 3 ( = k 3 • V 1 ) The scaling coefficient is different for area (k 2 ) and for volume (k 3 ). L Now, suppose you have a swing hanging from a tree, child size, and want to double its pro- portions (to support an adult). That is k = 2. To double the proportions means that the weight (or volume) is multiplied by 8 (k 3 =2 3 ). But, the strength of the rope is proportional to the rope’s cross-sectional area. So if k = 2, then the cross-sectional area is increased only 4-fold (2 2 ) when it has to be increased 8-fold to bear the weight of the larger adult. Simple geometric scaling doesn ± t always work!
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02_Allometry - Allometry page 1.01 RR Lew isometric example...

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