This Demonstration is based on idealized dynamics because it ignores the viscosity coefficients and temperature, which are of central concern to real glue. Instead, the movement of any glue particle simply obeys a geometrical rule. This has the advantage that we no longer need to consider the interaction among different glue particles.

We choose a

picture as a color source and extract color data from it. Then we apply the color data to a collection of random glue particles distributed within a cuboid region

. More specifically, first we partition the source picture into several subimages, then we calculate the mean color of each subimage, and finally we apply the mean colors to the

* *random glue particles in order. This gives a layer representative of the source picture, as long as the subimage's pixel value is small enough. The size of the cuboid region

runs from 1 to

, 1 to

, and 1 to

in the

,

, and

directions, where

,

, and

are determined by the magnitude of both the source picture and the subimage. It may take a long time to render with a small subimage setting because there can be a huge amount of particles with various colors, so do not set the subimage's pixel value to less than 10 in the source code unless you are using a powerful computer. You can use a different picture as a color source.

Here is the geometrical rule governing the movement of glue particles: if the ball's center is located at position

and a glue particle

, which is initially located at position

, moves to a new location

under the influence of

, then

(1)

,

where

is an adjustable parameter. We also apply a periodic framework setting in this Demonstration. So, if the ball

leaves the cuboid region

from its left boundary, it then enters from the right boundary at the same time. This periodic setting requires us to consider the influences of the eight corresponding balls

located in the eight surrounding congruent regions

that enclose

;

forms a bigger cuboid region that looks like

.

Any cuboid region

has length

, width

, and height

. If the center is located at position

where

, then

can be determined by

(2)

.

As a result, the glue particle

's new location

should be modified to

(3)

.

However, as the collection of

* *random glue particles normally contains several thousand particles and equation (3) greatly increases the calculation, we applied several simplified versions of equation (3) in the source code due to ball

's position within the cuboid region

, so that the summation in equation (3) contains only one, two, or four items.

(4)

In this way, we slightly sacrifice precision but gain remarkably on efficiency. Please refer to the program for more details about how to determine the position. Also, notice that

may belong to another

region.

Under the influence of the balls, a glue particle

will move from its initial position

to its new position

. If the color of this particle is described as

where

, then the force

between particle

and the ball

is defined as

(5)

,

where

is an adjustable parameter,

is treated as a vector, and the vector

refers to particle

's displacement caused by the existence of ball

(see equation (1), for example). It is obvious that

points to the sphere center of

, thus the ball

will only have translational motion. According to the periodic setting, we only need to calculate the resultant force acting on the ball

alone and track its motion step by step, since the locations of the other balls can be deduced from equation (2). (In fact, if we want to directly calculate the locations of all

, a 3×3

region is still insufficient.) During simulation, whenever

leaves the

region, we use the periodic setting to reset its position within

. In this way, it is always ensured that

and we only need to focus on the calculation regarding

alone throughout the whole simulation.

A cutoff setting for the resultant force calculation is also adopted in order to improve efficiency. That is, only those particles in the vicinity of ball

will be considered for the resultant force acting on

, because in this Demonstration it is ensured that

if

is not in the vicinity of

; thus, the force contribution of other particles can actually be ignored. More specifically, a glue particle

will be considered for the force calculation if

(6)

.

As all glue particles are initially randomly distributed, a test like equation (6) can be somewhat time consuming, so we also use a "floating filter" approximation to accelerate data processing; please refer to the source code for details. Based on the cutoff setting, the force

can be simply calculated as

(7)

for any particle

in the vicinity of

, because

's final position

is determined in advance in our source code.

When the ball

is located at the boundary of region

, parts of the ball will belong to other

regions, and the direct adoption of equation (7) requires us to consider particles belonging to other

regions as well. For example, if the position is at the bottom left, the force calculation involves particles belonging to region

as well as regions

,

, and

. But it will be much better if only the particles belonging to

are taken into consideration, since the number of particles in

can be large. This concern can be met by adopting the periodic setting—we can investigate the forces acting on the

,

, and

balls' corresponding parts instead, which now belong to

. In practice, we first need to take in the

region where those particles are interacting with

and

, then combine them in the correct order into a partitioned matrix. If we denote

as the matrix containing those particles' initial coordinates and

as the matrix containing those particles' final coordinates, then

provides all the data required by equation (7). The resultant force (represented as a red arrow) only exerts influence over the balls' motion, as all the glue particles' motion is governed by idealized (or fake) dynamics.

**Ball Lighting Specification**Like the cutoff setting adopted in equation (6), we define

and set those glue particles satisfying

as spotlight sources that aim at

with different half-angles, so that the light spots on the ball surface will have the same size. When the ball

is located at the boundaries of the region

, similar operation as mentioned above is applied.

If the ball lighting appears inverted when crossing

's boundaries, you need to search within the source code for

(*Reverse@*)VLPC and change them to

Reverse@VLPC (this occurs three times). This problem appears to depend on the operating system: on Windows XP,

VLPC yields the correct result, but on Windows 7, we need to use command

Reverse to modify

VLPC.

**Ball Radius Determination**As both the glue particle dynamics and the glue-ball interaction are idealized, the ball's size must be determined with caution so the Demonstration can appear valid. We mainly use the coordinates of those spotlight sources for radius calculation at every step, so the ball's size may vary at every step. (Actually, radius varies little and the naked eye can hardly notice the change.)

**2D Force Field Calculation**In order to better understand the ball's motion, we take about 1000 sample points evenly distributed within the middle section of the glue layer and calculate the corresponding force vectors (ignoring the

direction). This process may take several minutes to complete and it is not for real-time simulation, so we simply import the calculated result as an image.

A few videos of this Demonstration can be downloaded from my

personal page.