The Geometric Aspect of Ternary Forms Originally published in Werke Vol. VIII, pp. 348-50 by C. F. Gauss Translated from the German by Sky Jason Shields Let a point in space 0 be taken as origin. Let the transition [Übergang] from there to three other points P, P, P, which do not lie in a plane with the former, be t, t, t respectively, where, whenever no confusion is possible, the points P, P, P themselves can be represented by t, t, t. Further, let t, t be generally the product of the lengths shortest distance of the two lines t, t with the cosine of their inclination, etc. We have in general αt + α t + α t +..., βu + β u + β u +..., if the multiplication αt + α t + α t +... βu + β u + β u +... is carried out and t, u, t, u, t, u, t, u, t, u etc. is written instead of tu, tu, tu, t u, t u, etc. Every point in space can be represented by the trinomial xt + x t + x t The equation for all points which lie in a determined plane will then be λx + λ x + λ x = L where λ, λ, λ, L signify determined numbers. For a plane through the three points µt, µ t, µ t, we have If we write λµ = λ µ = λ µ = L. t, t = a, t, t = a, t, t = a, t, t = b, t, t = b, t, t = b and then a a bb = A, aa b b = A, aa b b = A, b b ab = B, bb a b = B, bb a b = B, D = aa a + 2bb b abb a b b a b b,
is perpendicular to T = At + B t + B t t and t T = B t + A t + Bt t and t T = B t + Bt + A t t and t, and in general, if λx + λ x + λ x = L is the equation of a plane, then the line λt + λ T + λ T will be perpendicular to it. L = a value of the form A, A, A B, B, B, if the indeterminates are set = λ, λ, λ. Then further at + b T + b T = Dt b T + a T + bt = Dt b T + bt + a T = Dt, and the lines t, t, t are perpendicular to the planes, whose equations are ax + b x + b x = Const. b x + a x + bx = Const. b x + bx + a x = Const. The doubled area of the triangle through the points mt, m t, m t is equal to the square root of the value of the form A, A, A F... B, B, B if the substitutions X = m m, X = mm, X = mm, are made, while the sixfold volume of the pyramid which they form with the zero point is = mm m D; therefore the perpendicular is D = F m, ; m, m T, T, T correspond to the form AD, A D, A D BD, B D, B D just as t, t, t to a, a, a b, b, b. The three roots of the equation 0 = p 3 ppa + a + a + pa + A + A D represent the squares of the three primary axes of an ellipsoid described in that parallelopiped to which the ternary positive form a, a, a A, A, A b, b, b with adjunct B, B, B and determin. = D 2
Connection of the Spatial Proportions [Raumverhältnisse] to a Given Tetrahedron Let 0,, 2, 3 be the four vertices, opposite faces, and perpendiculars [?]. There then arise for every point P of the space the four coordinates x, x, x, x, between which the relation x + x + x + x = 0 holds. That is to say that this signifies that x is the quotient, if the distance of the point P from a plane parallel to the plane 0 erected through the point M is divided by the perpendicular 0, etc. Then, in general P M 2 = xx 0 2 +xx 02 2 +xx 03 2 +x x 2 2 +x x 3 2 +x x 23 2 The Fundamental Theorem of Crystallization can be expressed most briefly in the following way: Between every five planes which appear therein, the following relation holds: If their normals to the surface of the sphere are 0,, 2, 3, 4, then the products sin02 sin304, sin03 sin204, sin203 sin04, are always in a rational proportion; if this proportion is α : β : γ, then β = α + γ. If the coordinates of the 5 points on the surface of the sphere are a b c a b c a b c a b c 0 0 then ab ba a b b a ab ba a b b a ab ba a b b a must be in a rational proportion. In general, let, 2, 3, 4, 5, the five points on the surface of the sphere, and 0 be the center; then, if 2 denotes the solid volume of the tetrahedron 0345 25 45 24 53 25 34 stand in a rational proportion as do 2 34 3 42 4 23 etc. 3
Transformation of the form 5, 5, 5,, determinant = 08 + + + + + + + + 0 0 + + + 0 + +2 + + + +2 + + + +2 + +2 +2 +2 + +2 +2 +2 + 7 7 7 7 7 7 36 08 8 8 8 +2 +2 +2 4 08 20 20 20 +4 +4 +4 6 08 29 29 29 +23 +23 +23 25 08 4 4 4 + + + 54 0 0 0 +7 +7 +7 26 240 240 240 +68 +68 +68 256 664 5 5 5 3 3 3 27 08 5 5 5 2 2 2 432 08 32 32 32 38 38 38 67500 08 If, generally, the original form is set = t, t, t u, u, u, and a derived form = T, T, T U, U, U then, T = 3t 2u U = t+ 2u 2, T = 2t+ 2u U = t+ 3u 3, T = 6t+ 0u U = 5t+ u 4, T = 9t+ 6u U = 8t+ 7u Via the substitution x = u + u 2u inverted 6u = x+ 3y + 2z y = u u 6u = x 3y + 2z z = u + u + u 6u = 2x+ 2z x z mod 3, x y mod 2 0 0 0 +7 +7 +7 26 5 5 5 54 7 7 7 7 7 7 6 08 52 52 52 23 23 23 33750 Calcite equiaxe 2 inverse contrastante mixte the form, 3, k 0, 0, 0 transfoms into 4+k, 4+k, 4+k k 2, k 2, k 2. In order to produce calcite, it is necessary to set k = 0.97303. If the complex values of the orthographic projection of three equally long and mutually perpendicular degrees [Graden] are a, b, c, then aa + bb + cc = 0 and we can set generally, p and qdenoting arbitrary complex numbers a = p qq pii, b = q qipii pi, c = qi ppi q. Calcite Calcium Carbonate, Calcareous Spar, Carbonate of Lime, Kalkspath, Kalkstein, Chaux carbonatée, also the double refracting Iceland Spar studied by Huyghens. Haüy says: The carbonate of lime, for example, takes according to circumstances the form of a rhombohedron rhomboidé, that of a regular hexagonal prism, that of a solid terminated by twelve scalenohedral triangles, that of a dodecahedron with pentagonal faces rhombohedron and hexagonal prism, etc. [http://www.minsocam.org/msa/collectors corner/arc/hauyv.htm] These are the various forms of the crystal examined by Gauss in the table. Chaux carbonatée is the very first entry in Haüy s Tableau méthodique des espèces minérales, which may be worth translating as an appendix in the sourcebook, though the entry is several pages long. It is available for free and in French on Google books. Perhaps Ben and Jason will look at it in Oakland in connection with the pentagramma fragments. SJS 2 The terms in this column are words coined by Haüy in order to describe different types of crystals formed by the same substance. See preceding footnote. SJS 4
Hexoctahedron [Hexakisoctaeder] Equation: px + qy + rz = Coordinates. γ 0 0 2. β+γ 3. α+β+γ β+γ 0 α+β+γ α+β+γ α < β < γ The sixfold contents of an elemental pyramid = γ β+γα+β+γ. Everything is inscribed on a sphere whose radius is = The double area of a triangle = αα+ββ+γγ γβ+γα+β+γ. αα+ββ+γγ. 2αα+β+γ 2 β+γα+β+γ. Edges 2 = ββ+γγ γβ+γ, 3 = α+β 2 +2γγ γα+β+γ, 2 3 = Cosine of the edge angle [? Cosinus Kanten Winkel] = 3 2 = Sine = = γ αα+ββ+γγ ββ+γγα+β 2 +2γγ Occuring Values αβ+ββ+γγ ββ+γγα+β 2 +2γγ The hexoctahedron is a form composed of forty-eight triangular faces, each of which cuts differently on all three crystallographic axes. There are several hexoctahedrons, which have varying ratios of intersection with the axes. A common hexocatahedron has for its parameter relations a, 3/2b, 3c, its symbol being 32. Other hexoctahedrons have the symbols 42, 53, 432, etc. it is to be noted the hexoctahedron is a form that may be considered as an octahedron, each face of which has been replaced by six others. It is to be recognized when in combination by the facts that there are six similar faces in each octant and that each face intercepts the three axes differently. Fig. 48 shows a simple hexoctahedron, Fig. 49 a combination of cube and hexoctahedron, and Fig. 50 a combinationof dodecahedron and hexoctahedron, and Fig. 5 a combination of dodecahedron, trapezohedron and hexoctahedron. [http://www.miningbasics.com/html/the isometric trisoctahedron.php] SJS 5
α β γ α β γ 7. 0. 0. Hexahedron 3 2.. 2. 2. Trisoctahedron 4 3. 0.. Rhombic Dodecahedron 4.. 2. 3. Hexoctahedron 6. 0.. 2 Tetrahexahedron 5. 2. 4 Hexoctahedron 0.. 3 2.. 3. 3. Trisoctahedron 0. 2. 3 4.. 3. 5 Hexoctahedron... Octahedron 2. 2. 3. 3. Trisoctahedron 5... 2 Trapezohedron 6 4. 2. 3. 4.?.. 3 3. 5.. 3 cube 4 The trisoctahedron is a form composed of twenty-four isosceles triangular faces, each of which intersects two of the crystallographic axes at unity and the third axis at some multiple. There are various trisosctahedrons the faces of which have different inclinations. A common trisoctahedron has for its parameters a, b, 2c, its symbol being 22. Other trisoctahedrons have the symbols 33, 44, 332, etc. It is to be noted that a tripezohedron has for its parameters a, b, 2c, its symbol being 22. Other tisoctahedrons have the symbols 33, 44, 332, etc. it is to be noted that the trisoctahedron, like the trapezohedron, is a form that may be conceived of as an octahedron, each face of which has been replaced by three others. Frequently it is spoken of as the trigonal trisoctahedron, the modyfin word indicating that its faces have each three edges and so differ from those of the trapezohedron. But when the word trisoctahedron is used alone it refers to this form. The following points would aid in its identification when it is found occurring in combination: the three similar faces in each octant; their relations to the axes, and the fact that the middle edges between them go toward the ends of the crystallographic axes. Fig. 43 shows the simple trosctahedron and Fig. 47 a combination of a trisoctahedron and an octahedron. It will be noted that the faces of the trisoctahedron bevel the edges of the octahedron. [http://www.miningbasics.com/html/the isometric trisoctahedron.php] Gauss notation seems to be the same as the modern notation, but reversed. SJS 5 The tetrahexahedron is a form composed of twenty-four isosceles triangular faces, each of which intersects one at unity, the second at some multiple, and is parallel to the third. There are a number of tetrahexahedrons which differ from each other in respect to the inclination of their faces. Perhaps the one most common in occurrence has the parameter relations a, 2 b, c, the symbol of which would be 20. The symbols of other forms are 30, 40, 320, etc. it is helpful to note that the tetrahexahedron, as its name indicates, is like a cube, the faces of which have been replaced by four others. [http://www.miningbasics.com/html/the isometric normal class.php] SJS 6 The trapezohedron is a form composed of twenty-four trapezium-shaped faces, each of which intersect one of the crystallographic axes at unity and the other two at equal multiples. There are various trapezohedrons with their faces having different angles of inclination. A common trapezohedron has for its parameters a, 2b, 2c, the symbol for which would be 2. The symbols for other trapezohedrons are 3,, 322, etc. it will be noted that a trapezohedron is an octahedral-like form and may be conceived of as an octahedron, each of the planes of which has been replaced by theree faces. Consequently it is sometimes called a tetragonal trisoctahedron. The qualifying word, tetragonal, is used to indicate that each of its faces has four edges and to distinguish it from the other trisoctahedral form, the description of which flows. Trapezohedron is the name, however, most commonly used. The following are aids to the recognition of the form when it occurs in combinations: the three similar faces to be found in each octant; the relations of each face to the axes; and the fact that the middle edges between the three faces in any one octant go toward points which are equidistant from the ends of the two adjacent crystallographic axes. It is to be noted that the faces of the common trapezohedron truncate the edges of the dodecahedron. ibid SJS 6