‘In the City Park’ is part of the collection ‘Views of a City’.

In ‘Views of a City’, I try to show different aspects of city life, including unpleasant ones such as toxic waste.

‘Atomic Egg’ is part of the collection ‘Views of a City’.

In ‘Views of a City’, I try to show different aspects of city life, including unpleasant ones such as toxic waste.

‘Wild Fighter’ is part of the collection ‘Twelfth Night’.

Depending on the region, three to twelve nights around Christmas time have a traditional significance. In German, they are called ‘Raunächte’ (Twelfth Night). According to a folk myth, during those nights the wild hunt is performed. The hunters may be either elves or fairies or the dead. Also ghosts and souls of the dead are being furloughed.

‘Strange World’ is part of the collection ‘Strange Worlds’.

‘Lime Plant’ is part of the collection ‘Strange Worlds’.

‘Instrument’ is part of the collection ‘Strange Worlds’.

‘Agricultural Crop Factory’ is part of the collection ‘Strange Worlds’.

A triangle with an edge lengths ratio 3:4:5 is rectangular because 3^2 + 4^2 = 5^2 fulfills the Pythagorean equation a^2 + b^2 = c^2 (Please note that x^2 signifies the second power of x).

The white object in the background is a downscaled model of the King’s Chamber in the Khufu Pyramid. Surprisingly, the red 3-4-5-triangle, a 12-knot drawstring, can be perfectly inscribed in the inside space.

The object in the foreground is a Spatial Pythagorean Stick Model. At the fixed vertical bar three marbles are countersunk, in the height of 7 units (red), 8 units (green) and 9 units (blue). Some complements are countersunk into the plane with respect to the possible integer Pythagorean solutions. The provided sticks with marked units have the lengths of integer diagonals.

In a spatial Cartesian coordinate system the distance d between a point P0 (0,0,0) and another point P1 (x,y,z) is given by the equation x^2 + y^2 + z^2 = d^2.

In accordance to the equation 12^2 + 20^2 + 9^2 = 25^2 the girl choosed the right stick to join the two blue marbles.

The edges of the pyramid are streched by red drawstrings. The black triangle gives the slope of the pyramid faces. The ratio of the horizontal edge of the triangle, a, to its vertical edge, b, is 11:14. The value for a is commonly specified by 115.19 m, the length of 220 Egyptian royal cubits. The result for the height b is 146.60 m, resp. 280 royal cubits.

The model shows the geometric way to construct the almost identical heights b(τ) with the golden ratio τ (τ = 1.61803…) and b(π) with the circular constant π (π = 3.14159…).

b(τ) = (τ^0.5) a = 146.52 m   

b(π) = 4 (π^-1) a = 146.66 m 

(please note: τ^0.5 = √τ and: π^-1 = 1/π )

The ancient methods of measurement were not accurate enough to find a difference between b, b(τ) and b(π). Perhaps this is why the pyramid was regarded as a fundamental geometric construction.

I. Viewed from the observation hole which is marked with the iron ring, the white ellipse fulfills exactly the black circle on the plexiglass window, i. e., the observation hole and the circle define a cone, with the ellipse as a cone section.

II. Johannes Kepler (1571-1630) constructed an ellipse on the base of intersection points given by the radius lines of two concentric circles. A horizontal straight line from the inner intersection point and a vertical straight line from the outer intersection point gives a cross-over point. Stringed together, these points are generating the ellipse. Please compare the crossing drawstrings in the upper left.

III: Using examples of points with integer coordinates it can be shown that the ellipse is a (horizontal) dilation of a circle.

IV. The ellipsograph, a stick movable on rectangularly mounted gliding channels, generates the same ellipse as a drawstring which is fixed in the focal points of the ellipse.

This mirror object, in which an inlaid hexagon appears 20 times, is very popular among children and adults.

The object consists of three mirrors in the form of  isosceles triangles with a base-to-height ratio of 1: 0.5 τ (τ = 0.5 (√5 + 1) = 1.61803 …). This mirror object is made of wood.

The three mirrors are joined together at their thighs so that they enclose a 72° angle. Therefore, the black stripes attached to the hexagons appear as black pentagons. There are also many other depositors available.

Some of the objects from this gallery are documented in detail in the exhibition “IMAGINARY - Form und Formel mathematischer Fantasie”, which was shown for the first time in Nuremberg in June 2016.

(www. imaginary.org/event/imaginary-in-nuremberg)

In the early 13th century Leonardo Fibonacci described a hypothetical rule to calculate the growth of a rabbit population. This rule generates a quasiperiodic sequence of young and adult rabbit pairs. Attaching the length S to the young pairs, and the length L to the adult pairs, the sequence corresponds perfectly to that one which is generated by the substitution method. 

The one dimensional (1D) quasi-cells in the upper part of the image generate the quasiperiodic sequence step by step. The successive growth is forced by sliding rulers (yellow). The rulers have a periodic average length. By connecting their end-marks, as shown in the image, the mechanisms of the quasi-cells force the growth of a perfect quasiperiodic sequence.

The 1D quasi-cells give the base for modelling flawless decagonal quasicrystalline growth by the algorithm of Quasiperiodic Succession. Detailed descriptions you will find in the files of the gallery Quasicrystalline Wickerwork. 

(www. imaginary.org/gallery/quasicrystalline-wickerwork)

For modelling growth of quasicrystals neither the matching rules of the Penrose rhombs nor the related covering rules of alternative overlapping unit-cells (Gummelt decagons) are strong enough to avoid mismatches.

The algorithm of the quasiperiodic succession also works step-by-step but it generates the growth of a flawless cartwheel order controlling each thick rhombus by a quasi-cell Q.

A quasi-cell Q has five sliding rulers with a periodic average length (yellow). The rulers of the start-cell have a starlike position. The big blue quasi-cell Q is related to the blue rhombus of the cartwheel order in the upper left. In this cell only one ruler is shown, movably guided on the respective red twin scale. The position of this ruler depends from the correlation with the parallel ruler of the start-cell in the lower left. Its end mark on the right allows to impart its position to the (not shown) ruler of the parallel green scale of the suivel-mounted black outlined plexiglass rhombus in the upper right. Its position is related to the green rhombus of the cartwheel order in the upper left.

The recursive formula consists of 50 scale value equations. The ten blocks in the outer columns correspond to the ten possible rhombus neighborhoods hj which are shown beside the columns. a, b, c, d and e are the values of the five double scales of a quasicell Q. The double scales are rotated by 36° to one another. Each thick rhombus can be assigned a quasicell Q.

The five values of the start cell Q0 are infinitesimally small, i. e. μ0 = 10^–n, with lim n → ∞. If they are entered in the central block, the formula leads to four valid neighbours, h1*, h2*, h4 and h1 (because their five respective values are valid) and six invalid neighbours (because one or more of their respective values are invalid). Please note that an asterisk is used here instead of an overline to indicate an inverse operation.

The valid values of Q01*, Q02*, Q04 and Q01are entered in the central block of four new formula sheets whereby the next generation of neighbours can be calculated. This recursiv scheme creates step by step an error-free quasiperiodic cartwheel order, as required for such a growth model.