A Geometry oíNutrition1 PARRY MOON AND DOMINA EBERLE SPENCER Electrical Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and Mathematics Department, University of Connecticut, Starrs, Connecticut 06268 ABSTRACT The nutritional value of a given sample of food may be specified by a nutrition holor in a fictitious 3-space. This holor possesses magnitude and character. In most cases, we are interested primarily in character, and this can be represented by a point in a 2-space (the nutrition triangle). The triangle allows visualization of re lations among foods and also the addition of nutritional values for a combination of foods. J. Nutr. 104: 1535-1542, 1974. INDEXING KEY WORDS geometry •mathematical model •nutrient patterns Downloaded from The worldwide importance of modern acter of a food may be specified by giving nutritional theory can hardly be overesti any two of the above coordinates. These mated. Nutrition calculations, however, coordinates can be plotted in a nutrition have remained in the arithmetic stage and triangle (fig. 1). have not utilized the possibilities of alge The diagram is advantageous because it braic and geometric formulation. In par shows at a glance the relations among jn.nutrition.org ticular, a geometrization of nutrition al foods: which food has the higher lipid lows one to visualize the relations among content, which has the more protein, etc. nutrients and thus to bypass much of the The triangle can be divided into regions. present routine computation, in accordance Sugars, syrups, and honey are almost pure with the principle that a picture is worth carbohydrate, so they are represented by by on November 25, 2009 a thousand words. a small region near the origin of coordi We wish to present, therefore, a possible nates. Butter, margarine, and cooking oils geometrization of nutrition, based on the are almost pure lipid and are represented specification in terms of protein, lipid, and near the top corner of the diagram. Most carbohydrate. Such a spécificationis ad cheeses are in the neighborhood of n1 = mittedly rudimentary; but it has proved to n2 = 0.5 with very little carbohydrate. be highly useful, and it can be supple Therefore, they constitute a region near mented by further requirements involving the middle of the right side of the triangle. vitamins, minerals, amino acids, and fatty Nuts lie in the upper middle part of the acids. triangle, with pecans near the top and The nutrition triangle. Chemical analysis peanuts at the bottom of this region. Vege of food yields values of the mass (kg) of tables and grains form a region near the protein, lipid (fat), and carbohydrate. bottom of the diagram, with the common Instead of using these absolute values characteristic of low lipid content. directly, however, it is usually more con Combinations. Perhaps the greatest im venient to employ relative values. Ex portance of the nutrition triangle is in de pressed as fractions of the total mass of termining the nutritional value of combina nutrients, we have 2 tions of foods. Any mixture of foods a and n1 = relative mass of protein ; Received for publication May 11, 1973. n2 = relative mass of lipid ; 1The term holor used In this paper was first used in 1963. Moon, P. & Spencer, D. E. (1963) A new n3 = relative mass of carbohydrate. mathematical representation of alternating currents. Tensor 14, 110-121. Because n1 + n2 + n3 = 1, only two of these 2Note that we are using superscripts Instead of subscripts, In accordance with standard tensor nota- quantities are independent. Thus the char- «ontor contravariantquantities. 1535 1536 PARRY MOON AND DOMINA EBERLE SPENCER I.O Downloaded from jn.nutrition.org \ by on November 25, 2009 VEGETABLES, GRAINS O N, 06 70 lO.QRR1ES0.2 0.3 10.4 0.5 I 0.6 07 0.8 0.9 IO ( X........ V^ ^HRorrni i SOYRFÕN Finiin fallUGAR BANANAS BROCCOLI SOYBEAN FLOUR Fig. 1 The nutrition triangle. The horizontal scale refers to protein, the vertical scale refers to lipid. b is represented in the diagram by a point (0.182, 0.560), indicating a cream between on the straight line ab. In figure 1 for ordinary milk and light cream. instance, different mixtures of buttermilk The amounts of the two ingredients can (0.402, 0.012) and butter (0.007, 0.989) be calculated by the methods described in are represented by the line between these Mixtures in 2-space. For example, sup terminal points. Reference to the triangle pose that we mix milk and flour in various shows that milk, light cream, and heavy cream fall on this locus. As another exam proportions, as represented by the line ab ple, consider ice cream (0.108, 0.337) ob of figure 2. A scale of tained by mixing a cream with sugar or kg flour honey (0, 0). The line passing through ff T— ~ these two points cuts the milk locus at kg milk A GEOMETRY OF NUTRITION 1537 04• a0.30.2 ALL MIXTURES O.In°.fiV«,^/FL7XI Downloaded from 10 Fig. 3 A mixture of three foods (a, b, c) gives a point within the shaded area. R/7MILKs/I0 O.I 0.2 • 0.3 the method described in Mixtures in 2- space. jn.nutrition.org Fig. 2 Any mixture of two foods (a and b) is represented by a straight line segment in the The nutrition holor. In the preceding nutrition triangle. sections, we have attempted to sketch some of the possibilities of nutritional geometry. In the remainder of the paper, can be established as shown in the dia we shall establish the mathematical basis by on November 25, 2009 gram. Thus, the exact proportions of flour for the method. Let us specify a given and milk can be determined for any point sample of food by three numbers repre- on the locus. Similarly, mixtures of three ingredients give all points within the shaded triangle, figure 3. This method can be extended to any number of ingredients, as suggested by figure 4. A convex polygon then includes all possible mixtures of the given constitu ents. Such graphical construction shows im mediately if a mixture of given ingredients can possibly result in a given proportion of nutrients. For example, suppose we take the point (0.100, 0.100) as our criterion. It is represented by the stars in figures 3 and 4. Because the shaded areas of figures 3 and 4 do not include the star, we can con clude immediately, without calculation, that no possible mixture of the proposed ingredients can satisfy the nutritional speci fication. By proper additions (in these cases, of sugar or fruit) however, the de Fig. 4 A mixture of any number of foods is sired point can be reached. Proportions of represented by a point within an area of the the ingredients can then be determined by nutrition triangle. 1538 PARRY MOON AND DOMINA EBERLE SPENCER .Z Downloaded from jn.nutrition.org by on November 25, 2009 Fig. 5 Nutrition 3-space. Each food is represented geometrically by a point in this space. senting kg of protein, kg of lipid, and kg merates : of carbohydrate. Evidently, these numbers may be considered as "components" or (tf 0. + (tf ')» = l(W)a + merates ( 1) of a nutrition holor N{: + (AT2)»,(N*)a + . (Eq. 2) kg protein Addition is commutative and associative. Multiplication by a scalar is meaningful: kg lipid kN{ = tf'fc = (kNl, kN2, kN*). (Eq. 3) kg carbohydrate For instance, if we have k identical sam ples of food, the total kg of protein is Nf = (AT1,N*, A"). (Eq. 1) equal to k times the protein of each sample. Equality of two holors (N*)tt and (W)& Lipid and carbohydrate could be deter mined in a similar manner. Note, however, is obtained if and only if corresponding that products of nutrition holors have no merates are equal : physical significance. There is nothing cor = (AT% responding to the familiar scalar and vector products of vector analysis ( 1). and Nutrition 3-space. Geometrically, Nl (N3). = (N3)b. may be considered as a point in a nutri tional 3-space (fig. 5). The axes represent Addition of two holors is obtained in pure protein, pure lipid, and pure carbo the usual way by adding corresponding hydrate, respectively. But for some pur- A GEOMETRY OF NUTRITION 1539 poses, it may be convenient to take other whose nutrition holors are (N')0 and (N')i,. axes, representing milk, butter, and cheese, Accordingthe mixturetoisEq.the2,"vector"the nutritionsum shownholor ofin for instance. Because the system is a linear one, we can transform from one set of axes figure 5. Note that this property continues to another, using a linear or affine trans to hold if we transform coordinates by formation ( 1) : changing scales along the axes and rotating N<r = A/'AT* axes in an affine transformation. (Eq. 4) Another property deals with multiplica The transformation matrix is tion by a scalar k. Because the same factor k appears in each merate, Eq. 3, a variation in k moves the point N' along a straight (Eq. 5) line radiating from the origin. Thus, one may establish a family of lines through The holor Nl is an affine invariant that is the origin, each line representing a differ independent of the coordinate system, ent kind of food. For each kind of food, the though of course its merates {N'} depend origin represents zero amount of that food, on the coordinates in which it is expressed. with amounts increasing linearly as we Nutrition space is an affine space (2-5), move outward from 0.