Math II (Несколько текстов для зачёта), страница 9

Then, for a divisor m of 18, the successive powers of p[sup m] take only 18/m values; for example, with m = 3, p[sup 3] = 8 and the powers of 8 modulo 19 are 1, 8, 7, 18, 11, 12. We can associate to this sequence the polynomial Z + Z[sup 8] + Z[sup 7] + Z[sup 18] + Z[sup 11] + Z[sup 12], which takes only three values when Z varies among the 19th roots of 1.

More generally, let n be a prime number and p a primitive nth root of 1 in the multiplicative group 1, 2, ..., n - 1. For a divisor m of n - 1, and d satisfying 0 </= d < (n - 1)/m, we define the sequence [m, d] as the polynomial [m, d] = Sigma[sub k = 0,1, ..., m - 1] Z[sup(sup[sup d + k(n - 1)/m)]mod n]. This notation is slightly different from Gauss's notation in [6].

As an example, with n = 19 and p = 2, we have the following sequences (the sequences of even length can be identified as the sums previously defined):

[18, O] = 1.2.3.4.5.6.7.8.9

= Z[sup 1] + Z[sup 2] + Z[sup 4] + Z[sup 8]

+ Z[sup 16] + Z[sup 13]+ Z[sup 7] + Z[sup 14]

+ Z[sup 9] + Z[sup 18] + Z[sup 17] + Z[sup 15]

+ Z[sup 11] + Z[sup 3] + Z[sup 6] + Z[sup 12]

+ Z[sup 5] + Z[sup 10],

[9, 0] = Z[sup 1] + Z[sup 4] + Z[sup 16] + Z [sup 7]

+ Z [sup 9] + Z[sup 17 ]+ Z[sup 11] + Z[sup 6]

+ Z[sup 5],

[9, 1] = Z[sup 2] + Z[sup 8] + Z[sup 13] + Z[sup 14]

+ Z[sup 18] + Z[sup 15]+ Z[sup 3] + Z[sup 12]

+ Z[sup 10],

[6, 0] = 1.7.8 = Z[sup 1] + Z[sup 8] + Z[sup 7] + Z[sup 18]

+ Z[sup 11] + Z[sup 12],

[6, 1] = 2.3.5 = Z[sup 2] + Z[sup 16] + Z[sup 14]

+ Z[sup 17] + Z[sup 3] + Z[sup 5],

[6, 2] = 4.6.9 = Z[sup 4] + Z[sup 13] + Z[sup 9]

+ Z[sup 15] + Z[sup 6] + Z[sup 10],

[3, O] = Z[sup 1] + Z[sup 7] + Z[sup 11],

[3, 1] = Z[sup 2] + Z[sup 14] + Z[sup 3],

[3, 2] = Z[sup 4] + Z[sup 9] + Z[sup 6],

[3, 3] = Z[sup 8] + Z[sup 18] + Z[sup 12],

[3, 4] = Z[sup 16] + Z[sup 17] + Z[sup 5],

[3, 5] = Z[sup 13] + Z[sup 15] + Z[sup 10],

[2, 0] = x[sub 1] = Z[sup 1] + Z[sup 18],

[2, 1] = x[sub 2] = Z[sup 2] + Z[sup 17],

[2, 2] = x[sub 4] = Z[sup 4] + Z[sup 17],

*[This character cannot be converted to ASCII text]

[2, 8] = x[sub 9] = Z[sup 9] + Z[sup 10],

[1, 0] = Z[sup 1],

[1, 1] = Z[sup 2],

*[This character cannot be converted to ASCII text]

[1, 17] = Z[sup 10].

When Z varies among the nth roots of 1, [m, d] takes only (n - 1)/m values, which are [m, 0], ..., [m, (n - 1)/ m - 1].

For n = 19, it follows that the sums 1.7.8, 2.3.5, and 4.6.9 are the zeros of H[sub 1] = X[sup 3] + X[sup 2] - 6X - 7. Let alpha be a zero of H[sub 1]; one of the three factors of Q[sub 19] in Q[alpha] is H[sub 2] = X[sup 3] alphaX[sup 2] + (alpha[sup 2] - 5)X + alpha[sup 2] - 6.

The zeros of H[sup 1] are found using the parabola 7Y[sup 2] - X + 6Y -1 = 0. Its axis is Y = -3/7 and it contains the points (-1, 0), (0, -1), and (15/4, 1/2). The zeros of H[sub 2] are found using the parabola (alpha[sup 2] - 6)Y[sup 2] + X + (alpha[sup 2] - 5)Y - alpha = 0. Its axis is Y = -alpha/2 - 1 and it contains the points (alpha, 0), (-alpha, alpha), and (alpha + 1, -1). The corresponding construction is given in Figure 11.

R[sub 37]

Q[sub 37] has degree 18, and we have the choice between three decompositions: 2Center dot3Center dot3, 3Center dot2Center dot3, and 3Center dot3Center dot2. We will present a construction using the first one. With the notations of the previous section, we take n = 37 and p = 2. The 2 sequences [18, *] are zeros of H[sub 1] = Z[sup 2] + Z - 9, the 6 sequences [6, *] are zeros of H[sub 2] = Z[sup 6] + Z[sup 5] - 15Z[sup 4] - 28Z[sup 3] + 15Z[sup 2] + 38Z - 1, and the 18 sequences [2, *] are zeros of Q[sub 37]. Let alpha be a zero of H[sub 1]; the zeros beta of H[sub 2] are zeros of Z[sup 3] - alphaZ[sup 2] + (-4 - 2alpha)Z + (7 + 2alpha); the zeros of Q[sub 37] are zeros of 11Z[sup 3] - 11betaZ[sup 2] + (4beta[sup 5] - 2beta[sup 4] - 57beta[sup 3] - 21beta[sup 2] + 97beta - 21)Z + (8beta[sup 5] - 4beta[sup 4] 114beta[sup 3] - 53beta[sup 2] + 194beta + 2). The coefficients of this last polynomials are not easy to construct. Fortunately, they can be expressed using linear combinations of longer sequences. We have the following relations, which allow us to construct R[sub 37]:

[18, 0], [18, 1] zeros of Z[sub 2] + Z - 9,

[18, 1] < [18, 0],

[6, 0], [6, 2], [6, 4] zeros of

Z[sup 3] - [18, 0] Z[sup 2] + (-4 - 2 [18, 0])Z

+ (7 + 2[18, 0]),

[6, 4] < [6, 5] < [6, 1] < [6, 3] < [6, 0] < [6, 2],

[2, 0], [2, 6], [2, 12] zeros of

Z[sup 3] - [6, 0] Z[sup 2] + ([6, 0] + [6, 4])Z

+ (-2 - [6, 1]),

[2, 17] < [2, 7] < [2, 4] < [2, 15] < [2, 13] < [2, 11]

< [2, 10] < [2, 12] < [2, 6] < [2, 16] < [2, 3]

< [2, 14]

< [2, 9] < [2, 5] < [2, 2] < [2, 8] < [2, 1]

< [2, 0].

The other relations are obtained by "shifting" the sequences: [m, d] is replaced by [m, (d + 1) mod (n - 1)/m].

In the proposed construction (Fig. 12), [18, *] are constructed using a circle (dark gray) of center (-1/2, - 4). [6, *] are constructed using two parabolas (medium gray). [2, *] are constructed using the six other parabolas (light gray).

R[sub 73] and R[sub 97]

As suggested by Bishop in [2], the construction can be restricted to give only one of the zeros of Q[sub n]. All the vertices of a regular polygon with a prime number of sides can be obtained by reflections from any pair of vertices. The reflections correspond to products of nth roots of 1.

For n = 73, we can choose p = 5 and the decomposition 36 = 2Center dot2Center dot3Center dot3. This leads to the following equations:

[36, 0], [36, 1] zeros of Z[sup 2] + Z - 18,

[18, 0], [18, 2] zeros of

Z[sup 2] - [36, 0] Z + 4 [36, 0] + 5

[36, 1],

[6, 0], [6, 4], [6, 8] zeros of

Z[sup 3] - [18, 0] Z[sup 2] - (2 + [18, 0]

+ [18, 3]) Z + 3 + 2 [18, 0] - 2 [18, 3],

[2, 0], [2, 12], [2, 24] zeros of

Z[sup 3] - [6, 0] Z[sup 2] + ([6, 0] + [6, 9])

Z - 3 - [6, 8].

Using these equations, "shifted" when needed, we can construct (Fig. 13) the following 15 values in 6 steps (each line corresponds to the construction of the zeros of a secondor third-degree polynomial):

[36, 0] = 3.772, [36, 1] = -4.772,

[18, 0] = 5.397, [18, 2] = -1.625,

[18, 1] = 0.047, [18, 3] = -4.819,

[6, 0] = 4.966, [6, 4] = -1.967, [6, 8] = 2.398,

[6, 1] = -1.580, [6, 5] = -1.538, [6, 9] = 3.166,

[2, 0] = 1.992, [2, 12] = 1.429, [2, 24] = 1.544,

For n = 97, we can use p = 5 and the decomposition 48 = 2Center dot2Center dot2Center dot2Center dot3, which corresponds to the equations

[48, 0] zeros of [48, 1] Z[sup 2] + Z - 24,

[24, 0], [24, 2] zeros of

Z[sup 2] - [48, 0]Z + 2[48, 0] - 5,

[12, 0], [12, 4] zeros of

Z[sup 2] - [24, 0] Z + 2 [48, 1] + 3 [24, 2] - [24, 1],

[6, 0], [6, 8] zeros of

Z[sup 2] - [12, 0] Z + [24, 1] + [12, 7],

[2, 0] [2, 16] [2, 32] zeros of

Z[sup 3] - [6, 0] Z[sup 2] + ([6, 0] + [6, 1l]) Z - 2 - [6, 2].

From these equations, we can deduce the following construction (Fig. 14) in 11 steps involving 23 values:

[48, 0] = 4.424, [48, 1] = -5.424,

[24, 0] = 1.189, [24, 2] = 3.234,

[24, 1] = 2.104, [24, 3] = -7.529,

[12, 0] = 2.493, [12, 4] = -1.303,

[12, 1] = 5.304, [12, 5] = -3.199,

[12, 2] = 0.079, [12, 6] = 3.155,

[12, 3] = -3.318, [12, 7] = -4.210,

[6, 0] = -0.666, [6, 8] = 3.159,

[6, 2] = 1.531, [6, 10] = -1.452,

[6, 3] = -0.441, [6, 11] = -2.877,

[2, 0] = 1.995, [2, 16] = -1.379, [2, 32] = -1.282.

Toward Automatic Construction

In the previous sections, we have presented more and more optimized ways to build the relations leading to constructions of the regular polygons. The final version can be summarized as follows.

Let n be an odd prime number of the form 2[sup a]3[sup b] + 1. For example, n = 2[sup 8] Center dot 3 + 1 = 769. The first step is to find p such that the powers of p modulo n generate the set {1, 2, ..., n -1}. For example, p = 11 for n = 769. Then, choose the decomposition of (n - 1)/2 into an ordered product of 2's and 3's. For example, for n = 97 = 2[sup 5]Center dot3 + 1, we have five choices: 2Center dot2Center dot2Center dot2Center dot3, 2Center dot2Center dot2Center dot3Center dot2, 2Center dot2Center dot3Center dot2Center dot2, 2<cd.>3Center dot2Center dot2Center dot2, and 3Center dot2Center dot2Center dot2Center dot2. In the general case, we have ([sup a + b - 1][sub b]) possible choices. It turns out to be more convenient to first solve second-degree polynomial equations.

The next step is to find the second- and third-degree polynomials whose zeros are the sequences corresponding to the previous decomposition. For n = 433 = 2[sup 4]Center dot3[sup 3] + 1, with p = 5 and the decomposition 216 = 2Center dot2Center dot2Center dot3Center dot3Center dot3, the lengths of the sequences are 216, 108, 54, 18, 6, and 2. We have to find the polynomials whose zeros are {[216, 0], [216, 1]}, {[108, 0], [108, 2]}, {[54, 0], [54, 4]}, {[18, 0], [18, 8], [18, 16]}, {[6, 0], [6, 24], [6, 48]}, and {[2, 0], [2, 72], [2, 144]}. The coefficients of these polynomials can be expressed as linear combinations of longer sequences, with integer coefficients. The first polynomial has integer coefficients.

The whole polygon can be deduced from the knowledge of only one of the sequences [2, *], say [2, 0]. We have to find the smallest set (or at least a reasonably small set) of sequences allowing the computation of [2, 0]. See the constructions of R[sub 73] and R[sub 97] for example (in the construction of R[sub 37], we constructed all the [2, *] sequences, without this simplification).

Finally, use the constructions of the zeros of secondand third-degree polynomials to build the successive sequences and, eventually, [2, 0]. This value gives a second vertex--we already have the point (2, 0)--of the polygon, which can be used to build all the others using reflections.

Conclusion

After recalling definitions and results about the constructibility of a geometric object, we have shown by more and more efficient methods how the works of Gauss, computer algebra systems (Maple), and dynamic geometry software (Cabri-Geometry, distributed by Texas Instruments) could be used together to construct regular polygons, using ruler, compass, and simple conics. In particular, we have given the list of small C[sub 2]-constructible polygons, and presented new C [sub 2]-constructions of the regular polygons with 19, 37, 73, and 97 sides.

The ancient Greeks gave precedence to constructions using only ruler and compass, not because they did not know about the other curves (they invented a number of mechanical devices drawing some algebraic curves of degrees 2, 3, 4, and more), but for the neatness, perfection of reasoning, and the simplicity of the shapes involved (circle and straight line).

Today's tools such as Cabri-Geometry enlarge the notion of geometric simplicity by allowing the manipulation of algebraic expressions (the sequences defined by Gauss) and complex geometric objects (the conic sections).

Some generalizations of the questions treated here may be considered:

1. What does the set of constructible numbers become if we

consider algebraic curves of higher degrees?

2. What is the asymptotic distribution of the primes of the form

2[sup a]3[sup b] + 1?

3. Can the C[sub 2]-constructions of the regular polygons be

fully automated?

4. Given n, what is the most efficient way of

C[sub 2]-constructing R[sub n], in terms of number of steps

and in terms of precision of the intersections involved

(avoiding intersection between near-tangent curves)?

CONSUL, THE EDUCATED MONKEY

Source: Mathematics Teacher, Apr2000, Vol. 93 Issue 4, p276, 4p, 7 diagrams

Author(s): Kolpas, Sidney J.; Massion, Gary R.

On 27 June 1916, The Educational Toy Manufacturing Company of Springfield, Massachusetts, patented "Consul," the Educated Monkey, a tin mathematical toy. According to the instructions accompanying the toy, the Educated Monkey was designed to

• teach the multiplication tables to 12s, associated elementary division, and associated elementary factoring and

• teach the addition tables to 12s and associated elementary subtraction.

When the monkey's feet are set to point at two numbers, its fingers locate their product. The photograph shows the left foot, from the reader's point of view, pointing to 4; the right foot, from the reader's point of view, pointing to 9; and the hands pointing to the product, 36. The entire multiplication table appears to form a 45 degrees-45 degrees-90 degrees triangle (fig. 1), and the triangle outlined by the monkey also appears to be a 45 degrees-45 degrees-90 degrees triangle, with the product 36 at the vertex of the right angle, as shown in figure 2. Figure 3 shows what appears to be another outlined 45 degrees-45 degrees-90 degrees triangle resulting from 7 x 11, with the product, 77, at the vertex of the right angle.

To square a number, the user sets the left foot to point to the number and sets the right foot to point to the symbol of a square (see fig. 1). The fingers then locate the square of the number.

To divide, the user sets one foot to point to the divisor and arranges the fingers to point to the dividend. The other foot then points to the quotient. To factor, the user makes the fingers point at a product. The feet then point to the factors.

For addition or subtraction, the toy comes with a cardboard addition table, shown in figure 4, that is slipped under the monkey and secured by paper fasteners to two slots on the plate of the toy. Addition and subtraction proceed in a manner similar to multiplication and division.