2124:
1302:
2119:{\displaystyle {\begin{aligned}a'&=\left\|{\begin{pmatrix}-a\\K-b\\-c\end{pmatrix}}-\left({\begin{pmatrix}-a\\K-b\\-c\end{pmatrix}}\cdot {\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\right){\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\right\|\\&=\left\|{\begin{pmatrix}-a\\K-b\\-c\end{pmatrix}}-\left(0+{\frac {K-b}{\sqrt {2}}}+{\frac {c}{\sqrt {2}}}\right){\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\right\|\\&=\left\|{\begin{pmatrix}-a\\K-b-{\frac {K-b+c}{2}}\\-c+{\frac {K-b+c}{2}}\end{pmatrix}}\right\|=\left\|{\begin{pmatrix}-a\\{\frac {K-b-c}{2}}\\{\frac {K-b-c}{2}}\end{pmatrix}}\right\|\\&={\sqrt {{(-a)}^{2}+{\left({\frac {K-b-c}{2}}\right)}^{2}+{\left({\frac {K-b-c}{2}}\right)}^{2}}}={\sqrt {a^{2}+{\frac {{(K-b-c)}^{2}}{2}}}}\,.\end{aligned}}}
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323:. The first method is to use a diagram grid consisting of lines parallel to the triangle edges. A parallel to a side of the triangle is the locus of points constant in the component situated in the vertex opposed to the side. Each component is 100% in a corner of the triangle and 0% at the edge opposite it, decreasing linearly with increasing distance (perpendicular to the opposite edge) from this corner. By drawing parallel lines at regular intervals between the zero line and the corner, fine divisions can be established for easy estimation.
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387:
911:
122:
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435:
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1286:{\displaystyle \mathbf {s} ={\begin{pmatrix}0\\K\\0\end{pmatrix}}\quad {\text{and}}\quad \mathbf {\hat {n}} ={\frac {{\begin{pmatrix}0\\K\\0\end{pmatrix}}-{\begin{pmatrix}0\\0\\K\end{pmatrix}}}{\left\|{\begin{pmatrix}0\\K\\0\end{pmatrix}}-{\begin{pmatrix}0\\0\\K\end{pmatrix}}\right\|}}={\frac {\begin{pmatrix}0\\K\\-K\end{pmatrix}}{\sqrt {0^{2}+K^{2}+{(-K)}^{2}}}}={\begin{pmatrix}0\\{\frac {1}{\sqrt {2}}}\\-{\frac {1}{\sqrt {2}}}\end{pmatrix}}\,.}
2723:
25:
351:. The third method does not require the drawing of perpendicular or parallel lines. Straight lines are drawn from each corner, through the point of interest, to the opposite side of the triangle. The lengths of these lines, as well as the lengths of the segments between the point and the corresponding sides, are measured individually. The ratio of the measured lines then gives the component value as a fraction of 100%.
355:
A displacement along a parallel line (grid line) preserves the sum of two values, while motion along a perpendicular line increases (or decreases) the two values an equal amount, each half of the decrease (increase) of the third value. Motion along a line through a corner preserves the ratio of the
824:
2301:
2712:
2395:
716:
1307:
896:
333:. For diagrams that do not possess grid lines, the easiest way to determine the values is to determine the shortest (i.e. perpendicular) distances from the point of interest to each of the three sides. By
2852:
2153:
410:
2878:
2572:
422:
2596:
398:
2455:
260:
for all substances being graphed, any one variable is not independent of the others, so only two variables must be known to find a sample's point on the graph: for instance,
2320:
470:
3519:
2915:
2838:
2824:
2808:
This sample has the same proportion of clay as Sample 2, but the proportions of silt and sand are swapped; the plot is reflected about its vertical axis.
819:{\displaystyle \left\|(\mathbf {s} -\mathbf {p} )-{\bigl (}(\mathbf {s} -\mathbf {p} )\cdot \mathbf {\hat {n}} {\bigr )}\mathbf {\hat {n}} \right\|\,.}
2895:
386:
374:
2791:
The proportion of sand is 30% as in Sample 1, but as the proportion of silt rises by 40%, the proportion of clay decreases correspondingly.
3221:
839:
458:
362:
196:, and other physical sciences to show the compositions of systems composed of three species. Ternary plots are tools for analyzing
2774:
Because clay and silt together make up 70% of this sample, the proportion of sand must be 30% for the components to sum to 100%.
2731:
2866:
2296:{\displaystyle a'={\sqrt {a^{2}+{\frac {{(a+b+c-b-c)}^{2}}{2}}}}={\sqrt {a^{2}+{\frac {a^{2}}{2}}}}=a{\sqrt {\frac {3}{2}}}\,.}
2966:
2956:
166:
89:
2526:
289:
is that three variables can be conveniently plotted in a two-dimensional graph. Ternary plots can also be used to create
61:
2707:{\displaystyle \left({\frac {1}{2}}\cdot {\frac {2b+c}{a+b+c}},{\frac {\sqrt {3}}{2}}\cdot {\frac {c}{a+b+c}}\right)\,.}
446:
3266:
108:
2400:
This shows that the distance of the point from the respective lines is linearly proportional to the original values
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46:
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42:
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57:
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3386:
312:
There are three equivalent methods that can be used to determine the values of a point on the plot:
3371:
3306:
2986:
3326:
2946:
35:
3424:
3419:
3276:
2722:
2390:{\displaystyle b'=b{\sqrt {\frac {3}{2}}}\quad {\text{and}}\quad c'=c{\sqrt {\frac {3}{2}}}\,.}
3524:
3504:
297:
3238:
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2951:
2910:
2486:
are useful for plotting points in the triangle. Consider an equilateral ternary plot where
615:
334:
286:
177:
176:
to a constant. It graphically depicts the ratios of the three variables as positions in an
126:
3147:. Loughborough University – Department of Geography / Resources Gateway home > Tri-plot
82:
8:
3351:
3321:
338:
282:—it is possible to graph the combinations of all three variables in only two dimensions.
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197:
181:
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3079:
Boyd, S. and
Vandenberghe, L., 2004. Convex optimization. Cambridge university press.
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3251:
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2884:
Ternary triangle plot of soil types sand clay and silt programmed with
Mathematica
278:. Because the three numerical values cannot vary independently—there are only two
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3434:
3366:
3331:
3316:
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3296:
3261:
3181:"Data Analysis in the Geosciences – Ternary Diagrams developed in the R language"
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519:
169:
2737:
This example shows how this works for a hypothetical set of three soil samples:
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3256:
3246:
416:
Figure 5. An example ternary diagram, showing increments along the second axis.
293:
by outlining the composition regions on the plot where different phases exist.
121:
428:
Figure 6. An example ternary diagram, showing increments along the third axis.
404:
Figure 4. An example ternary diagram, showing increments along the first axis.
3574:
3544:
3381:
3341:
3286:
2920:
488:
296:
The values of a point on a ternary plot correspond (up to a constant) to its
290:
3396:
3058:
Karl Tuyls, "An evolutionary game-theoretic analysis of poker strategies",
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2727:
2424:
Analogue on a
Cartesian grid by adding lines of slope −1. The scale of the
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212:
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Being dependent on the first two, the intersect is on the 30% sand line
193:
189:
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3559:
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3429:
3199:
3002:
891:{\displaystyle \mathbf {p} ={\begin{pmatrix}a\\b\\c\end{pmatrix}}\,.}
185:
173:
24:
3376:
3014:
2976:
3161:"Ternary Plot Generator – Quickly create ternary diagrams on line"
3479:
3281:
3271:
392:
Figure 3. An example ternary diagram, without any points plotted.
220:
130:
242:. Usually, this constant is represented as 1.0 or 100%. Because
3160:
3301:
3127:. Science Education Resource Center (SERC) Carleton College
481:
492:
Derivation of a ternary plot from
Cartesian coordinates
476:
Unlabeled triangle plot with major and minor grid lines
337:, the distances (or the ratios of the distances to the
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207:, a triangle plot of genotype frequencies is called a
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2567:{\textstyle ({\frac {1}{2}},{\frac {\sqrt {3}}{2}}),}
2323:
2156:
1305:
914:
842:
719:
226:
In a ternary plot, the values of the three variables
3520:
List of boiling and freezing information of solvents
285:
The advantage of using a ternary plot for depicting
2995:
2889:
307:
49:. Unsourced material may be challenged and removed.
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2566:
2449:
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2118:
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452:Figure 8. Indication of how the three axes work.
3215:
3141:"Tri-plot: Ternary diagram plotting software"
790:
749:
464:Unlabeled triangle plot with major grid lines
618:view. The triangle, viewed face-on, appears
3222:
3208:
2415:
1296:Using the perpendicular distance formula,
2700:
2383:
2289:
2108:
1279:
884:
812:
599:is restricted to the triangle bounded by
109:Learn how and when to remove this message
2721:
2419:
614:In (3), the axes are rotated to give an
487:
120:
3178:
3088:
2732:United States Department of Agriculture
3573:
3229:
3082:
2816:
3203:
3121:"Excel Template for Ternary Diagrams"
3033:
2967:List of information graphics software
2957:Barycentric coordinates (mathematics)
482:Derivation from Cartesian coordinates
559:is restricted to a plane containing
47:adding citations to reliable sources
18:
3089:Vaughan, Will (September 5, 2010).
2450:{\textstyle {\frac {1}{\sqrt {2}}}}
341:) give the value of each component.
16:Barycentric plot on three variables
13:
2465:axes. The cross denotes the point
14:
3597:
3113:
2972:Earth sciences graphics software
2926:UIGS Classification diagram for
2890:List of notable ternary diagrams
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2865:
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2837:
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798:
779:
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758:
737:
729:
469:
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409:
397:
385:
373:
361:
308:Reading values on a ternary plot
23:
2982:Origin (data analysis software)
2356:
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962:
956:
200:in the three-dimensional case.
34:needs additional citations for
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808:
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688:is a unit vector) and a point
1:
3020:
2306:Similar calculation on lines
380:Figure 2. Intersection method
3510:Inorganic nonaqueous solvent
3068:10.1016/j.entcom.2009.09.002
440:Figure 7. Empty ternary plot
129:, showing which mixtures of
7:
2940:
2858:Plotting Sample 1 (step 3):
2844:Plotting Sample 1 (step 2):
2830:Plotting Sample 1 (step 1):
238:must sum to some constant,
10:
3602:
3495:Acid dissociation constant
2807:
2790:
2773:
2717:
3478:
3410:
3340:
3237:
2936:diagram by particle sizes
625:In (4), the distances of
595:each cannot be negative,
368:Figure 1. Altitude method
172:on three variables which
3179:Holland, Steven (2016).
2989:has a dedicated package
2923:, used in hydrochemistry
2872:Plotting all the samples
3460:Solubility table (data)
3327:Apparent molar property
3187:. University of Georgia
3060:Entertainment Computing
2947:Apparent molar property
2416:Plotting a ternary plot
555:(a positive constant),
302:barycentric coordinates
219:, it is often called a
3425:Total dissolved solids
3420:Solubility equilibrium
3345:and related quantities
2846:Find the 20% silt line
2832:Find the 50% clay line
2734:
2708:
2568:
2480:
2451:
2391:
2297:
2120:
1287:
892:
820:
696:perpendicular distance
493:
142:
3525:Partition coefficient
3505:Polar aprotic solvent
3040:mathworld.wolfram.com
2725:
2709:
2569:
2484:Cartesian coordinates
2452:
2423:
2392:
2298:
2121:
1288:
893:
821:
491:
298:trilinear coordinates
287:chemical compositions
124:
3440:Enthalpy of solution
3367:Volume concentration
3362:Number concentration
3097:on December 20, 2010
2911:Flammability diagram
2896:Chromaticity diagram
2597:
2527:
2432:
2321:
2154:
1303:
912:
840:
829:In this case, point
717:
498:Figure (1) shows an
178:equilateral triangle
127:flammability diagram
43:improve this article
3352:Molar concentration
3322:Dilution (equation)
3165:www.ternaryplot.com
3034:Weisstein, Eric W.
2817:Plotting the points
518:in a 3-dimensional
349:intersection method
217:convex optimization
205:population genetics
3392:Isotopic abundance
3357:Mass concentration
3231:Chemical solutions
2962:Compositional data
2916:Jensen cation plot
2901:de Finetti diagram
2735:
2730:triangle from the
2704:
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1918:
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1008:
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878:
816:
500:oblique projection
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327:Perpendicular line
280:degrees of freedom
209:de Finetti diagram
198:compositional data
182:physical chemistry
143:
3568:
3567:
3125:serc.carleton.edu
3036:"Ternary Diagram"
2952:Viviani's theorem
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804:
785:
356:other two values.
335:Viviani's theorem
264:must be equal to
119:
118:
111:
93:
3593:
3465:Solubility chart
3292:Phase separation
3252:Aqueous solution
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3093:. Archived from
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3050:
3049:
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3010:Project triangle
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2015:
2004:
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1525:
1516:
1510:
1506:
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1123:
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1012:
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919:
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832:
825:
823:
822:
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811:
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797:
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787:
786:
778:
769:
761:
753:
752:
740:
732:
709:
703:
693:
687:
682:in vector form (
681:
662:, respectively.
661:
654:
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628:
610:
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602:
598:
594:
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582:
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558:
554:
534:, respectively.
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389:
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180:. It is used in
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107:
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100:
94:
92:
51:
27:
19:
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3595:
3594:
3592:
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3590:
3571:
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3569:
3564:
3474:
3435:Solvation shell
3406:
3344:
3336:
3332:Miscibility gap
3317:Serial dilution
3312:Supersaturation
3262:Buffer solution
3233:
3228:
3190:
3188:
3169:
3167:
3159:
3150:
3148:
3145:www.lboro.ac.uk
3139:
3130:
3128:
3119:
3116:
3111:
3110:
3100:
3098:
3091:"Ternary plots"
3087:
3083:
3078:
3074:
3057:
3053:
3044:
3042:
3032:
3028:
3023:
2943:
2928:Ultramafic rock
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2656:
2634:
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2604:
2600:
2598:
2595:
2594:
2575:
2574:and the triple
2546:
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1443:
1442:
1432:
1431:
1421:
1420:
1411:
1410:
1398:
1397:
1384:
1383:
1382:
1378:
1368:
1367:
1358:
1357:
1345:
1344:
1331:
1330:
1329:
1325:
1318:
1310:
1306:
1304:
1301:
1300:
1272:
1271:
1259:
1253:
1252:
1240:
1237:
1236:
1226:
1225:
1213:
1199:
1198:
1189:
1185:
1176:
1172:
1165:
1164:
1155:
1154:
1148:
1147:
1137:
1135:
1118:
1117:
1111:
1110:
1104:
1103:
1093:
1092:
1082:
1081:
1075:
1074:
1068:
1067:
1057:
1056:
1055:
1051:
1043:
1042:
1036:
1035:
1029:
1028:
1018:
1017:
1007:
1006:
1000:
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993:
992:
982:
981:
980:
978:
964:
963:
957:
949:
948:
942:
941:
935:
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924:
923:
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913:
910:
909:
902:
877:
876:
870:
869:
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862:
852:
851:
843:
841:
838:
837:
830:
796:
795:
789:
788:
777:
776:
765:
757:
748:
747:
736:
728:
724:
720:
718:
715:
714:
705:
699:
689:
683:
666:
656:
649:
642:
641:are denoted by
638:
634:
630:
626:
608:
604:
600:
596:
592:
588:
584:
576:
568:
560:
556:
538:
531:
527:
523:
520:Cartesian space
503:
484:
477:
474:
465:
462:
453:
450:
441:
438:
429:
426:
417:
414:
405:
402:
393:
390:
381:
378:
369:
366:
339:triangle height
331:altitude method
310:
265:
261:
243:
239:
235:
231:
227:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3599:
3589:
3588:
3583:
3566:
3565:
3563:
3562:
3557:
3552:
3547:
3542:
3537:
3532:
3527:
3522:
3517:
3512:
3507:
3502:
3500:Protic solvent
3497:
3492:
3484:
3482:
3476:
3475:
3473:
3472:
3467:
3462:
3457:
3452:
3447:
3445:Lattice energy
3442:
3437:
3432:
3427:
3422:
3416:
3414:
3408:
3407:
3405:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3369:
3364:
3359:
3354:
3348:
3346:
3338:
3337:
3335:
3334:
3329:
3324:
3319:
3314:
3309:
3304:
3299:
3297:Eutectic point
3294:
3289:
3284:
3279:
3274:
3269:
3264:
3259:
3257:Solid solution
3254:
3249:
3247:Ideal solution
3243:
3241:
3235:
3234:
3227:
3226:
3219:
3212:
3204:
3198:
3197:
3185:strata.uga.edu
3176:
3157:
3137:
3115:
3114:External links
3112:
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1597:
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1571:
1570:
1568:
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1528:
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1513:
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1224:
1216:
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1128:
1122:
1116:
1113:
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1102:
1099:
1098:
1096:
1091:
1086:
1080:
1077:
1076:
1073:
1070:
1069:
1066:
1063:
1062:
1060:
1054:
1047:
1041:
1038:
1037:
1034:
1031:
1030:
1027:
1024:
1023:
1021:
1016:
1011:
1005:
1002:
1001:
998:
995:
994:
991:
988:
987:
985:
977:
971:
968:
953:
947:
944:
943:
940:
937:
936:
933:
930:
929:
927:
922:
918:
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872:
871:
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865:
864:
861:
858:
857:
855:
850:
846:
827:
826:
815:
810:
803:
800:
792:
784:
781:
775:
772:
768:
764:
760:
756:
751:
746:
743:
739:
735:
731:
727:
723:
496:
495:
483:
480:
479:
478:
475:
468:
466:
463:
456:
454:
451:
444:
442:
439:
432:
430:
427:
420:
418:
415:
408:
406:
403:
396:
394:
391:
384:
382:
379:
372:
370:
367:
360:
353:
352:
342:
324:
309:
306:
291:phase diagrams
163:Gibbs triangle
117:
116:
58:"Ternary plot"
31:
29:
22:
15:
9:
6:
4:
3:
2:
3598:
3587:
3584:
3582:
3579:
3578:
3576:
3561:
3558:
3556:
3553:
3551:
3548:
3546:
3543:
3541:
3538:
3536:
3533:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3490:
3486:
3485:
3483:
3481:
3477:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3446:
3443:
3441:
3438:
3436:
3433:
3431:
3428:
3426:
3423:
3421:
3418:
3417:
3415:
3413:
3409:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3387:Mass fraction
3385:
3383:
3382:Mole fraction
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3349:
3347:
3343:
3342:Concentration
3339:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3287:Phase diagram
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3267:Flory–Huggins
3265:
3263:
3260:
3258:
3255:
3253:
3250:
3248:
3245:
3244:
3242:
3240:
3236:
3232:
3225:
3220:
3218:
3213:
3211:
3206:
3205:
3202:
3186:
3182:
3177:
3166:
3162:
3158:
3146:
3142:
3138:
3126:
3122:
3118:
3117:
3096:
3092:
3085:
3076:
3069:
3065:
3062:January 2009
3061:
3055:
3041:
3037:
3030:
3026:
3016:
3013:
3011:
3008:
3004:
3001:
2998:
2994:
2993:
2988:
2985:
2983:
2980:
2978:
2975:
2973:
2970:
2969:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2944:
2935:
2931:
2929:
2925:
2922:
2921:Piper diagram
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2902:
2899:
2897:
2894:
2893:
2880:
2875:
2868:
2863:
2854:
2849:
2840:
2835:
2826:
2821:
2820:
2804:
2801:
2798:
2795:
2794:
2787:
2784:
2781:
2778:
2777:
2770:
2767:
2764:
2761:
2760:
2756:
2753:
2750:
2747:
2744:
2743:
2740:
2739:
2738:
2733:
2729:
2728:soil textural
2724:
2701:
2696:
2689:
2686:
2683:
2680:
2677:
2673:
2668:
2663:
2659:
2653:
2647:
2644:
2641:
2638:
2635:
2630:
2627:
2624:
2621:
2615:
2610:
2607:
2601:
2593:
2592:
2591:
2587:
2583:
2579:
2561:
2553:
2549:
2543:
2538:
2535:
2520:
2509:
2502:
2498:
2493:is placed at
2490:
2485:
2477:
2473:
2469:
2464:
2460:
2441:
2437:
2427:
2422:
2413:
2384:
2377:
2374:
2368:
2365:
2361:
2358:
2344:
2341:
2335:
2332:
2328:
2325:
2317:
2316:
2315:
2290:
2283:
2280:
2274:
2271:
2264:
2259:
2255:
2249:
2244:
2240:
2234:
2227:
2222:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2180:
2175:
2171:
2165:
2161:
2158:
2150:
2149:
2148:
2145:
2141:
2137:
2133:
2129:Substituting
2109:
2101:
2096:
2088:
2085:
2082:
2079:
2076:
2066:
2061:
2057:
2051:
2044:
2038:
2033:
2029:
2026:
2023:
2020:
2017:
2011:
2005:
2000:
1994:
1989:
1985:
1982:
1979:
1976:
1973:
1967:
1961:
1956:
1948:
1945:
1935:
1933:
1919:
1911:
1907:
1904:
1901:
1898:
1895:
1883:
1879:
1876:
1873:
1870:
1867:
1857:
1854:
1848:
1839:
1830:
1822:
1818:
1815:
1812:
1809:
1806:
1800:
1797:
1794:
1785:
1781:
1778:
1775:
1772:
1769:
1763:
1760:
1757:
1754:
1747:
1744:
1738:
1729:
1727:
1712:
1703:
1699:
1694:
1684:
1680:
1671:
1665:
1659:
1652:
1648:
1643:
1637:
1632:
1629:
1626:
1620:
1617:
1613:
1609:
1604:
1598:
1595:
1588:
1585:
1582:
1575:
1572:
1566:
1556:
1554:
1539:
1530:
1526:
1521:
1511:
1507:
1498:
1492:
1486:
1480:
1471:
1467:
1462:
1452:
1448:
1439:
1433:
1428:
1423:
1417:
1414:
1407:
1404:
1401:
1394:
1391:
1385:
1379:
1375:
1370:
1364:
1361:
1354:
1351:
1348:
1341:
1338:
1332:
1322:
1320:
1314:
1311:
1299:
1298:
1297:
1280:
1274:
1265:
1261:
1256:
1246:
1242:
1233:
1227:
1222:
1214:
1206:
1203:
1195:
1190:
1186:
1182:
1177:
1173:
1167:
1161:
1158:
1151:
1144:
1138:
1132:
1120:
1114:
1107:
1100:
1094:
1089:
1084:
1078:
1071:
1064:
1058:
1045:
1039:
1032:
1025:
1019:
1014:
1009:
1003:
996:
989:
983:
975:
951:
945:
938:
931:
925:
920:
908:
907:
906:
885:
879:
873:
866:
859:
853:
848:
836:
835:
834:
813:
773:
762:
744:
733:
713:
712:
711:
708:
702:
697:
692:
686:
680:
677:
673:
669:
665:For any line
663:
659:
652:
645:
623:
621:
617:
612:
611:, as in (2).
580:
572:
564:
553:
549:
545:
541:
535:
521:
515:
511:
507:
501:
490:
486:
485:
472:
467:
460:
455:
448:
443:
436:
431:
424:
419:
412:
407:
400:
395:
388:
383:
376:
371:
364:
359:
358:
357:
350:
346:
343:
340:
336:
332:
328:
325:
322:
318:
317:Parallel line
315:
314:
313:
305:
303:
299:
294:
292:
288:
283:
281:
276:
272:
268:
258:
254:
250:
246:
224:
222:
218:
214:
210:
206:
201:
199:
195:
191:
187:
183:
179:
175:
171:
168:
164:
160:
156:
155:triangle plot
152:
151:ternary graph
148:
140:
136:
132:
128:
123:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3450:Raoult's law
3402:Ternary plot
3401:
3397:Mixing ratio
3189:. Retrieved
3184:
3168:. Retrieved
3164:
3149:. Retrieved
3144:
3129:. Retrieved
3124:
3101:September 7,
3099:. Retrieved
3095:the original
3084:
3075:
3059:
3054:
3043:. Retrieved
3039:
3029:
2990:
2934:Soil texture
2736:
2726:A colorized
2585:
2581:
2577:
2518:
2507:
2500:
2496:
2488:
2482:
2475:
2471:
2467:
2462:
2458:
2457:that of the
2425:
2399:
2305:
2143:
2139:
2135:
2131:
2128:
1295:
900:
828:
706:
700:
690:
684:
678:
675:
671:
667:
664:
657:
650:
643:
624:
613:
578:
570:
562:
551:
547:
543:
539:
536:
513:
509:
505:
497:
354:
348:
344:
330:
326:
320:
316:
311:
295:
284:
274:
270:
266:
256:
252:
248:
244:
225:
202:
162:
159:simplex plot
158:
154:
150:
147:ternary plot
146:
144:
139:nitrogen gas
137:, and inert
105:
99:January 2014
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
3555:Lyonium ion
3470:Miscibility
3455:Henry's law
3070:, p. 9
2906:Dalitz plot
629:from lines
620:equilateral
345:Corner line
321:grid method
213:game theory
167:barycentric
3575:Categories
3550:Amphiphile
3545:Lipophilic
3540:Hydrophile
3535:Hydrophobe
3412:Solubility
3307:Saturation
3277:Suspension
3045:2021-06-05
3021:References
522:with axes
194:metallurgy
190:mineralogy
135:oxygen gas
125:A ternary
69:newspapers
3586:Triangles
3560:Lyate ion
3515:Solvation
3430:Solvation
3372:Normality
3003:Sigmaplot
2669:⋅
2616:⋅
2503:) = (0,0)
2212:−
2206:−
2086:−
2080:−
2027:−
2021:−
1983:−
1977:−
1946:−
1905:−
1899:−
1877:−
1871:−
1855:−
1810:−
1795:−
1773:−
1764:−
1758:−
1745:−
1695:−
1630:−
1610:−
1596:−
1586:−
1573:−
1522:−
1463:−
1429:⋅
1415:−
1405:−
1392:−
1376:−
1362:−
1352:−
1339:−
1257:−
1204:−
1159:−
1090:−
1015:−
970:^
802:^
783:^
774:⋅
763:−
745:−
734:−
616:isometric
502:of point
186:petrology
141:will burn
3581:Diagrams
3530:Polarity
3489:Category
3377:Molality
3239:Solution
3015:Trilemma
2977:IGOR Pro
2941:See also
2796:Sample 3
2779:Sample 2
2762:Sample 1
2428:axis is
2362:′
2329:′
2162:′
1925:‖
1844:‖
1836:‖
1734:‖
1719:‖
1561:‖
1546:‖
1327:‖
1315:′
1127:‖
1053:‖
809:‖
722:‖
3480:Solvent
3282:Colloid
3272:Mixture
2992:ternary
2718:Example
2516:. Then
685:n̂
679:n̂
221:simplex
131:methane
83:scholar
3191:14 May
3170:14 May
3151:14 May
3131:14 May
2757:Notes
2745:Sample
2521:= 100%
2510:= 100%
2491:= 100%
2314:gives
833:is at
694:, the
577:C(0,0,
234:, and
223:plot.
85:
78:
71:
64:
56:
3302:Alloy
2932:USDA
2514:(1,0)
901:Line
698:from
583:. If
565:,0,0)
211:. In
165:is a
161:, or
90:JSTOR
76:books
3193:2020
3172:2020
3153:2020
3133:2020
3103:2010
2997:CRAN
2754:Sand
2751:Silt
2748:Clay
2505:and
2461:and
2408:and
2310:and
905:has
655:and
637:and
607:and
591:and
575:and
569:B(0,
530:and
215:and
170:plot
62:news
3064:doi
2805:60%
2802:30%
2799:10%
2788:30%
2785:60%
2782:10%
2771:30%
2768:20%
2765:50%
2590:is
2523:is
2512:at
2353:and
959:and
710:is
704:to
573:,0)
537:If
347:or
329:or
319:or
300:or
203:In
174:sum
45:by
3577::
3183:.
3163:.
3143:.
3123:.
3038:.
2474:=
2470:=
2412:.
2404:,
2312:AB
2308:AC
2147:,
2142:+
2138:+
2134:=
903:BC
674:+
670:=
648:,
639:AB
635:AC
633:,
631:BC
622:.
603:,
587:,
567:,
561:A(
550:=
546:+
542:+
526:,
504:P(
304:.
273:−
269:−
255:=
251:+
247:+
230:,
192:,
188:,
184:,
157:,
153:,
149:,
145:A
133:,
3491:)
3487:(
3223:e
3216:t
3209:v
3195:.
3174:.
3155:.
3135:.
3105:.
3066::
3048:.
2999:)
2987:R
2702:.
2697:)
2690:c
2687:+
2684:b
2681:+
2678:a
2674:c
2664:2
2660:3
2654:,
2648:c
2645:+
2642:b
2639:+
2636:a
2631:c
2628:+
2625:b
2622:2
2611:2
2608:1
2602:(
2588:)
2586:c
2584:,
2582:b
2580:,
2578:a
2576:(
2562:,
2559:)
2554:2
2550:3
2544:,
2539:2
2536:1
2531:(
2519:c
2508:b
2501:y
2499:,
2497:x
2495:(
2489:a
2478:.
2476:c
2472:b
2468:a
2463:b
2459:a
2442:2
2438:1
2426:c
2410:c
2406:b
2402:a
2385:.
2378:2
2375:3
2369:c
2366:=
2359:c
2345:2
2342:3
2336:b
2333:=
2326:b
2291:.
2284:2
2281:3
2275:a
2272:=
2265:2
2260:2
2256:a
2250:+
2245:2
2241:a
2235:=
2228:2
2223:2
2218:)
2215:c
2209:b
2203:c
2200:+
2197:b
2194:+
2191:a
2188:(
2181:+
2176:2
2172:a
2166:=
2159:a
2144:c
2140:b
2136:a
2132:K
2110:.
2102:2
2097:2
2092:)
2089:c
2083:b
2077:K
2074:(
2067:+
2062:2
2058:a
2052:=
2045:2
2039:)
2034:2
2030:c
2024:b
2018:K
2012:(
2006:+
2001:2
1995:)
1990:2
1986:c
1980:b
1974:K
1968:(
1962:+
1957:2
1952:)
1949:a
1943:(
1936:=
1920:)
1912:2
1908:c
1902:b
1896:K
1884:2
1880:c
1874:b
1868:K
1858:a
1849:(
1840:=
1831:)
1823:2
1819:c
1816:+
1813:b
1807:K
1801:+
1798:c
1786:2
1782:c
1779:+
1776:b
1770:K
1761:b
1755:K
1748:a
1739:(
1730:=
1713:)
1704:2
1700:1
1685:2
1681:1
1672:0
1666:(
1660:)
1653:2
1649:c
1644:+
1638:2
1633:b
1627:K
1621:+
1618:0
1614:(
1605:)
1599:c
1589:b
1583:K
1576:a
1567:(
1557:=
1540:)
1531:2
1527:1
1512:2
1508:1
1499:0
1493:(
1487:)
1481:)
1472:2
1468:1
1453:2
1449:1
1440:0
1434:(
1424:)
1418:c
1408:b
1402:K
1395:a
1386:(
1380:(
1371:)
1365:c
1355:b
1349:K
1342:a
1333:(
1323:=
1312:a
1281:.
1275:)
1266:2
1262:1
1247:2
1243:1
1234:0
1228:(
1223:=
1215:2
1210:)
1207:K
1201:(
1196:+
1191:2
1187:K
1183:+
1178:2
1174:0
1168:)
1162:K
1152:K
1145:0
1139:(
1133:=
1121:)
1115:K
1108:0
1101:0
1095:(
1085:)
1079:0
1072:K
1065:0
1059:(
1046:)
1040:K
1033:0
1026:0
1020:(
1010:)
1004:0
997:K
990:0
984:(
976:=
967:n
952:)
946:0
939:K
932:0
926:(
921:=
917:s
886:.
880:)
874:c
867:b
860:a
854:(
849:=
845:p
831:P
814:.
799:n
791:)
780:n
771:)
767:p
759:s
755:(
750:(
742:)
738:p
730:s
726:(
707:l
701:p
691:p
676:t
672:s
668:l
660:′
658:c
653:′
651:b
646:′
644:a
627:P
609:C
605:B
601:A
597:P
593:c
589:b
585:a
581:)
579:K
571:K
563:K
557:P
552:K
548:c
544:b
540:a
532:c
528:b
524:a
516:)
514:c
512:,
510:b
508:,
506:a
275:b
271:a
267:K
262:c
257:K
253:c
249:b
245:a
240:K
236:c
232:b
228:a
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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