# Voronoi's Diagrams in the Natural Science and Art

Nikolas is interested in the history of mathematics and its application in studying phenomena of the natural science and arts

## From Nothing

In many countries there are Institutes of scientific history. The forgotten discoveries, researches may unexpectedly arise and be very successful in solving theoretical and practical problems of our days. As known, the history of science is notable for a sudden appearance of certain knowledge "from nothing". So quick action anybody can not suppose.

In accordance with the French philosopher Michel Foucault, brokenness, as in our case with Voronoi diagrams, is one of essential characteristics in the history of science

## The Main Periods of Life

Georgy Theodosiyovych Voronoi(1868-1908) is famous for his fruitful contributions to number theory. They are those involving quadratic forms and Voronoi tessellations.

He was born and grew up in the town of Zhuravka in the Ukraine. His father was director in Berdiansk Man's Gymnasium, where George was pupil till the 5th grade. Then, G.Voronoi studied at St. Petersburg University under the supervision of Andrey Markov

In 1894, he became a professor of pure mathematics at the University of Warsaw. In the period of his career he published six large memoirs and six short papers. All works were so significant that they left a deep trace in modern number theory.

Together with Minkowski, he can remain as the founder of the Geometry of Numbers.

## Example from the Application in the Crystallography

One way of using Voronoi diagrams is by growing crystals.

1. Some crystals, all growing at the same rate, and at the same time, get many growing circles. As these circles meet, straight line boundaries appear among them. Each crystal will exactly fill up the Voronoi region of its point of origin. In reality, crystals start growing at different times.
2. Even if they still grow at the same rate, if they start at different times, they will no longer meet in straight lines. Instead, they will meet in hyperbolic segments. The diagram shows the "weighted, as additive, Voronoi diagram". It is defined like the usual Voronoi diagram. In contrast, each site has a weight, and you add it to the usual Euclidean distance.
3. Now suppose that all the crystals start forming at the same time, but grow at different rates. You get what is called the "weighted, as multiplicative, Voronoi diagram". Once again, each site gives a weight, but when you measure the distance to a site, you multiply its weight. As a result, the boundaries among different regions are segments of circles.

## From Ukraine to Corea. Center of History

• Voronoi Diagram Research Center has started on July 1, 2003. It happened due to the Creative Research Initiative program from Korean government. The staff was working on various applications of geometric modeling and manipulation. On the other hand, the center targeted to develop computing the Voronoi diagram.
• There were its higher dimensional spheres. Also, it was identifying applications in molecular biology and material science. The structural molecular biology is more appropriate applications for this theory. Moreover, the structural molecular biology field has long been requiring this theory. Terminating Initiative Program on June 30, 2007, Research Center continued as National Research Lab. Aims were more practical than those of Creative Research Initiative.
• The lab focused its effort on bio-molecules in the three-dimensional space. Also, the body completed the theory of the quasi-triangulation and the beta-complex.
• At the same time, the center implemented it into a software. Then, application of software is based on powerful capabilities outperforming commercial software.

## Diagrams and Art

It considers Voronoi diagrams as a set of points. His general drawing concept can create beautiful, different, and sometimes unexpected images in many arts.

This tessellation appeared due to the Voronoi diagram of colored rings. It is in hexagonal pattern

## The True Value

The achievements of Georgy Voronoi made great influence on his contemporaries. Experts recognized him as one of the brightest talents in the number theory, geometry. It was late XIX and early XX centuries.

The true value of his scientific heritage appeared itself only in our days. There was in connection with the computerization. Moreover, many other modern branches of science use it. Also, it is important to mark that Voronoi diagram is universal. You can find it as in arts as in science

## Certain Applications of Voronoi Diagram

ScienceApplicationsMeaning

Geometry of Numbers, Voronoi Diagrams

Archaeology and Anthropology

Identify regions under different neolithic places, ceremonial centers, forts, or chiefdoms

Astronomy

Describe the regions of gravitational influence of the Sun and other stars, identify clusters of galaxies

Biology, Ecology, Forestry

Model and study plant areas

Cartography

Combine satellite picture into whole maps

Chemistry and crystallography

Model alloy structures

## Sometimes, Mathematica may be Funny

For each question, choose the best answer. The answer key is below.

1. What phenomenon can you drive around on the space?
• point
• line
• circle
2. What volume does the patch of color represent?
• triangle
• circle
• area
3. Where is thje area located?
• closer to the locator
• further from the locator
• among the locators