In this paper, a problem of packing hexagonal and dodecagonal sensors in a circular container is considered. We concentrate on the sensor manufacturing application, where sensors need to be produced from a circular wafer with maximal silicon efficiency (SE) and minimal number of sensor cuts. Also, a specific application is considered when produced sensors need to cover the circular area of interest with the largest packing efficiency (PE). Even though packing problems are common in many fields of research, not many authors concentrate on packing polygons of known dimensions into a circular shape to optimize a certain objective. We revisit this problem by using some well-known formulations concerning regular hexagons. We provide mathematical expressions to formulate the difference in efficiency between regular and semiregular tessellations. It is well-known that semiregular tessellation will cause larger silicon waste, but it is important to formulate the ratio between the two, as it affects the sensor production cost. The reason why we have replaced the “perfect” regular tessellation with semiregular one is the need to provide spacings at the sensor vertices for placing mechanical apertures in the design of the new CMS detector. Archimedean
Packing is a common problem in many different fields, such as computer science, design, manufacturing, and engineering. It consists of embedding smaller shapes into a larger one called container [
There are applications with different optimization goals, like minimizing the size of the container, or maximizing the number of arranged inner components [
Packing problem can be mapped to the field of sensor manufacturing [
When producing a single sensor, it is known that SE is maximal if hexagon shape is used with respect to triangle or square [
Nowadays, it is possible to use methods of forming a semiconductor die that does not require applying only the single straight lines. Hence, dies having various nonrectangular shapes can be formed, increasing the wafer yield [
Many applications require production and use of these abovementioned polygonal sensors, especially hexagons [
A hexagonal silicon sensor for CMS [
With these vertex cuts, irregular dodecagon sensor shape needs to be produced, where a regular dodecagon is obtained with maximal cut applied. SE is expected to be larger when a single sensor is produced, but with 12 straight cuts applied on the wafer in the manufacturing process. The question that is posed in this study is can we construct a sensor whose geometrical shape can be as efficient in terms of minimal number of six cuts such as using a regular hexagon? Possibly, for these irregular hexagonal sensors, no vertex cuts are needed since they are provided by default in the tessellation process.
Naturally, applying the vertex cuts and using the irregular sensors both decrease SE when
The paper is organized as follows. First, related work is summarized in Section
Basic requirement when producing sensors is cost reduction by improving wafer productivity. This is defined as the fraction of the used wafer area to the total wafer area [
Many studies are done on maximizing wafer efficiency, mostly concentrating on square or rectangular sensors. A method for optimizing number of square cells that geometrically fit on a circular wafer and maximizing wafer yield is developed in [
Unlike in the field of sensor manufacturing where the region of interest is a circular silicon wafer or a container, the circular CMS detector represents region of interest that needs to be efficiently covered. Also, in the field of sensor networks, sensor positions are crucial in the form of a hexagonal grid to obtain better coverage efficiency and to cover the sensing area more efficiently [
There are many research problems on packing and various shapes of packed objects and containers are studied. Using rectangular or square container approximation is provided in the literature, concentrating mostly on circular packing problems. Litvinchev et al. [
Authors in [
Despite using a rectangular container model as in the previous researches, it can be adjusted to handle some other geometrical shape such as polygon or circle. Provided solutions can support the simple change of different container approximations [
In this paper, we adopt the idea of using a regular grid methodology to provide solutions of a packing problem. We present mathematical constructions to calculate the packing density of embedding regular and irregular hexagons and dodecagons in a circular container. Based on the literature search, none of the papers deal with these specific types of cutting polygonal sensor shapes. Hence, we address that specific type of packing problem when irregular hexagons and dodecagons are packed into circular container and we derive PE formulas.
We define four different sensor shapes produced from a circular silicon wafer, that are presented in Figure
Defined sensor geometrical shapes: (a) regular hexagon, (b) regular dodecagon, (c) irregular dodecagon, and (d) irregular convex hexagon.
We provide a study in two directions; first, we analyze SE when
A tiling or a tessellation is covering a flat surface using one or more tiles without any overlaps or gaps. Although the used tiles can have arbitrary shape, we focus on regular polygons (hexagons and dodecagons). Concerning the mathematical tiling context, edge is a boundary which separates the two tiles, and vertex is an endpoint of an edge.
We say that a tiling of the plane is an
Tiling is monohedral if all the tiles are of the same shape and size. Otherwise, it can be dihedral, trihedral, etc. Regular tiling is an edge-to-edge tiling by using congruent regular polygons. We could say it is a subtype of monohedral tiling. Uniform tiling is the tessellation of the plane by regular polygons with the restriction of being vertex-transitive. There are just three regular uniform tiling types given in Figure
Three regular tessellations (adjusted from [
It is mathematically trivial to prove tessellation with regular hexagons. The sum of all angles in one vertex is 360°. The sum of the inner polygon angles is
Archimedean or semiregular tessellation uses more than one type of regular polygon with the restriction of being vertex-transitive.
We show the proof of the simplest possibility which allows usage of two different types of regular polygons in the tessellation. The following equation represents the fact that single vertex is a meeting point for
Archimedean
Sensors are defined by four geometrical shapes, as shown in Figure
We use the following approach in solving the above problems. First, we start from a well-known regular hexagon tessellation and revisit the covering an area of interest without voids or gaps. We use formulas to find solutions for this general case when regular hexagons are packed in the circular container. Next, since this “perfect” regular tessellation is not satisfactory in our case, not providing spacings for mechanical apertures, we derive Archimedean
It is well known that PE is decreased when semiregular tessellation is applied with respect to the regular tessellation. The ratio is defined already for the problem of packing circles into a squared container [
Because of the flexibility in choosing the size of the triangular spacings with lowering the side of a regular dodecagon, we move from the regular dodecagons to the irregular ones (with 6 symmetry axes). We study the efficiency when packing them into a circular container. Irregular dodecagons can provide us the desired PE, but with larger number of cuts when producing a single sensor (number of cuts is 12, compared to a regular hexagon that needs only 6 cuts in the production). Hence, we examine the possibility of the sensor to remain hexagonal in shape. We construct a sensor that is in a form of an irregular symmetric hexagon and we pack them in the circular area.
To minimize the number of needed cuts of the sensor and together with provided triangular spacings, we propose semiregular tessellation with irregular hexagons (with 3 symmetry axes) that is not edge-to edge. Our goal is to compare our proposed packing with other shapes semiregularly tessellating the plane, considering that triangles must be cut out by default, so that we can have a fair comparison. Finally, we compare PE for each sensor type.
The questions that need to be answered with this study are formed by two aspects: Producing Which one is the most efficient? Is producing an irregular hexagon cost-effective? What is the SE ratio between regular and semiregular tessellation? Is producing Packing sensors of known dimensions in the circular area of interest and calculating PE which represents coverage efficiency: Flexible-size container in which we always pack the same number of sensors Which sensor shape requires the container to be the smallest? Which sensor shape reduces the unused packed space in the container and increases PE when area of interest is covered? Which sensor shape enables that larger area of interest can be covered?
Let us consider the regular hexagon tessellation with the coordinate system set like it is shown in Figure
Regular hexagon rings.
Number of hexagons in a ring is equal to 6. If we denote the number of hexagons in
By using the same assumption (
Again, we reformulate the problem: for a given circle of radius
If we compare
Using the same logic with the notation like in the previous regular hexagon tessellation case, we derive problem solutions on Archimedean
Regular dodecagon rings.
If
Applying the same logic as in regular tessellation, using (
Again, we use the same logic as in the regular hexagon case. Assumption (
If we take regular hexagon and starting from each vertex, we move along the side for the same distance
(a) Irregular dodecagon, cut from regular hexagon. (b) For maximum x-cut value, dodecagon is regular.
Let us now define the irregular dodecagon rings (Figure
Irregular dodecagon rings.
Triangular spacing is equilateral triangle with side
Let us consider an irregular convex hexagon
(a) Irregular hexagon
The hexagonal shape below is adjusted from [
We refer to irregular hexagon from Figure
Irregular hexagon. (a)
Triangular spacing is equilateral triangle with side
For
If we compare areas of the regular and irregular hexagon with the same circumscribed circle radius, we get
We apply the same logic in describing irregular hexagon rings (Figure
Irregular hexagon rings.
Parameters
In the previous discussion, we have narrowed the choice on irregular hexagons that have two different sides (
The area of an irregular hexagon
PE is calculated with using every sensor geometrical shape, approximating SE in the sensor production cost and PE which represents coverage efficiency when packing sensors of known dimensions in the circular area of interest. In the former context, we separate the following cases: A single sensor ( Sensors are produced from the fixed wafer size and they are packed into a circular container whose size is flexible. It is defined with the condition that always a fixed number of sensors
When a single sensor is produced, one must consider the useful wafer area available in the manufacturing process, as shown in Figure
Straight lines applied in single sensor production.
When the proposed irregular hexagon
SE for irregular hexagon
Comparison of regular and irregular hexagon area.
Irregular hexagon
For a single sensor in the selected class of irregular hexagons (or for a specific
Since regular dodecagon is the closest to the circular shape (Figure
When
As we can see in Table
Regular Hexagon and Regular Dodecagon for 8’’ wafer (R=100 mm).
Regular hexagon | Regular dodecagon | ||||||
---|---|---|---|---|---|---|---|
k | | a [mm] | | SE | a [mm] | | SE |
0 | 1 | 100 | 25980.8 | | 51.8 | 30004.01 | |
1 | 7 | 37.8 | 3711.5 | | 17.8 | 3544.9 | |
2 | 19 | 22.94 | 1367.4 | | 10.7 | 1282.7 | |
3 | 37 | 16.4 | 702.2 | | 7.7 | 655.3 | |
4 | 61 | 12.8 | 425.9 | | 5.95 | 396.7 | |
5 | 91 | 10.5 | 285.5 | | 4.9 | 265.6 | |
6 | 127 | 8.9 | 204.6 | | 4.12 | 190.2 | |
7 | 169 | 7.7 | 153.7 | | 3.6 | 142.9 | |
8 | 217 | 6.8 | 119.7 | | 3.15 | 111.2 | |
9 | 271 | 6.07 | 95.9 | | 2.8 | 89.1 | |
10 | 331 | 5.5 | 78.5 | | 2.6 | 72.9 | |
Let us compare several semiregular tessellations with irregular dodecagons used, where smaller triangles are cut than in the case of Archimedean tessellation. We can see in Figure
SE for semiregular tessellations with dodecagons (8’’ wafer, R=100 mm).
Maximal waste is obtained for the Archimedean semiregular tessellation, since maximal triangles are cut out from the wafer, causing the largest silicon waste.
When producing
SE for irregular hexagon
SE of the for every
For lower number of sensor rings
We can conclude that one can produce
According to (
This section will obtain the same PE results as the efficiency in the previous one expressed in a form of SE. Namely, when sensor size is fixed, and the area of the container is adjusted accordingly, this is the same as when we have a fixed region of interest and we adjust the sensor size. The reason is that we always use the cantered tessellation approach defined by the number of sensor rings
When sensors are produced from the fixed-size wafers and we pack them in the circular container, we can discuss the size of the container that is required. Namely, the size of the container is flexible because we want to pack sensors of fixed dimensions with the centred tessellation mechanism and the restriction that all packed sensors are whole, there are no partial sensors. The size of a container is always defined such that a fixed and tessellated number of sensors with the valid restriction (
When the size of the container is defined by the number of sensor rings that are packed inside, PE is the smallest for regular dodecagon or Archimedean semiregular tessellation. This means that leaving largest triangles unused in the sensor vertices causes the smallest coverage of the circular surface. On the other hand, we get the largest spacing area for mechanical apertures, but individual sensors are produced with maximal number of cuts. Using larger irregular dodecagons (like in the previous section) can reduce the unused packed space because triangular spacings are smaller, which increases PE towards the most efficient regular hexagonal tessellation used (83%). However, triangular spacings are smaller and again we need to apply maximal number of cuts in a single sensor production.
Concluded from Figure
We can compare the size of the container that we need depending on how many sensor rings do we pack inside. Sensor is produced from a fixed-size wafer and example for 6’’ sensors is given in Figure
Size of the flexible container used for packing (6’’ wafer, R=75 mm) compared to a regular hexagon. (a) Irregular hexagons, (b) irregular dodecagons.
Irregular dodecagons packed in the container cause the fact that its size can be smaller, and closer to the size of the container when regular hexagons are used. In this case, the smallest container can be used for packing since regular hexagons have the largest PE. On the other hand, since the area of an irregular hexagon is smaller than the area of a regular hexagon, smaller container is needed for the same number of sensors that are packed inside (Figure
Regular tessellation enables area to be covered with no voids or gaps, ensuring the constant
Producing such dodecagonal sensors (
The number of cuts needed in producing a single sensor is increased in double when dodecagon sensor shape is used compared with hexagon. Therefore, in this paper, we answered the question if we can use some other more efficient sensor shape that keeps the
Considering a specific application when produced sensors need to cover the circular area of interest with the satisfactory
As future work, we will observe a more realistic case where a problem of packing fixed-size sensors in a fixed-size container is studied. When sensors are produced from a fixed-size wafer and we aim to pack them into a fixed container, we first need to calculate how many of them can be fit inside. Then, based on a requirement (
The data used to support the findings of this study is generated based on the formulas included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.