[color=#0000ff][i][color=#0000ff][i][color=#999999]This activity belongs to the GeoGebra book [url=https://www.geogebra.org/m/mes4bgft]The Domain of the Time[/url].[/color][/i][/color][/i][/color][br][br]If, in the previous activity, we add a third point and make the pursued ones also pursuers, we obtain the well-known [i]mice problem[/i].[br][br]Now there are three points, [color=#ff7700]M[/color], [color=#ff7700]N[/color] and [color=#ff7700]P[/color], representing the mice, located respectively at the initial positions [color=#9900ff]A[/color], [color=#9900ff]B[/color] and [color=#9900ff]C[/color], which are the vertices of an equilateral triangle. Each point has the same constant speed (1 m/s), but each mouse is always directed toward its neighbor: [color=#ff7700]M[/color] moves toward [color=#ff7700]N[/color], [color=#ff7700]N[/color] moves toward [color=#ff7700]P[/color], and [color=#ff7700]P[/color] moves toward [color=#ff7700]M[/color].[br][br]The curve described by each mouse is one of the most common curves in nature, found in both shells and galaxies: the [i]logarithmic spiral[/i]. It doesn't matter whether there are 3 or more mice located at the vertices of a regular polygon; in any case, they will trace out a logarithmic spiral, also called [i]equiangular [/i]due to the constant angle formed by the tangent at any point with the line connecting it to the center (a property shared with the circle). In the case of the three mice, this constant angle is 30º.[br][br]You can modify the initial positions [color=#9900ff]A[/color] and [color=#9900ff]B[/color]. You can also download the construction and add more mice to adapt it to regular polygons with more sides: it's very simple!

[b]SCRIPT FOR SLIDER anima[/b][br][br][color=#cc0000][color=#cc0000]# Calculate the elapsed seconds dt; add one second if t1(1) < tt[/color][/color][br][color=#999999]SetValue(tt, t1(1))[br]SetValue(t1, First(GetTime(), 3))[br]SetValue(dt, (t1(1) < tt) + (t1(1) − tt)/1000)[/color][br][br][color=#cc0000]# Move M, N and P and stop the animation when N and M are close enough[/color][br][color=#999999][color=#999999]SetValue[/color](M, M + dt vM)[br][color=#999999]SetValue[/color](N, N + dt vN)[/color][color=#0000ff][br]SetValue(P, P + dt vP)[/color][color=#999999][br]StartAnimation(anima, abs(N [color=#999999]−[/color] M) > (x(Corner(2) [color=#999999]−[/color] Corner(1))/200))[/color][br][br][br][br][br][br][color=#999999][color=#999999][color=#0000ff][color=#0000ff][color=#999999][color=#999999]Author of the activity and GeoGebra construction: [/color][/color][/color][color=#0000ff][color=#999999][color=#999999][url=https://www.geogebra.org/u/rafael]Rafael Losada[/url].[/color][/color][/color][/color][/color][/color]