Skip to main content# New Calculator Design for Efficient Interface based on the Circular Group Approach

# 1 Introduction

## 1.1 Tradition in Design

## 1.2 The Research Question

# 2 Hypothesis

# 3 Theory

## 3.1 Binary Operators

## 3.2 Analogy to DVD Technology

## 3.3 Force Reactance in Key-Pressing

## 3.4 Analogy to Courier Logistics

## 3.5 Architecture of the Circular Group Design

## 3.6 Explanation of Groups 2 and 3

## 3.7 Overview of an End-Product

## 3.8 Note on Key Physical Qualities

# 4 Analysis

# 5 Results

## 5.1 Distance between keys

## 5.2 Time between keys

## 5.3 Comparison between designs

## 5.4 Examples

## 5.5 Hundred Calculations with Minus

## 5.6 Hundred Calculations with Other Operators

## 5.7 Overall Observation

## 5.8 University Example

# 6 Future Work

# 7 Conclusion

# 8 References

Published onMar 08, 2021

New Calculator Design for Efficient Interface based on the Circular Group Approach

During the second half of the 20^{th} century, calculators evolved at an astounding rate, focusing on reduction of size, weight, power consumption while at the same time achieving increased computational capability (in particular from 1960-70s, as suggested by Figure 1 constructed with data from various sources [1-5]), till the present day. During this time, the initial keyboard design did not changed significantly, and in some aspects, none at all (for example the number stacking bottom to top in a square matrix format, with the decimal and 0 key placed below). With time, additional functions were stacked on top of the matrix progressively (as seen on the right of Figure 1).

This lack of development in key design organization has represented a loss of handling efficiency for user, which when multiplied by millions of people computing for decades, signifies an important opportunity for improvement. One perspective of why this happened, is that all the design effort was engaged in the daunting technological challenge of creating a functional calculator, and miniaturizing it for economy scale commercialization.

History proved the aforementioned design effort to be a great success, as modern calculators now fit into pockets and have ever increasing computational and memory power, with the latest addition being advanced graphing capability in a color display, even in three-dimensional space [6]. This implied shifting from a mechanical architecture, to electrical-mechanical hybrid to fully electronic – whilst preserving the matrix key design - to achieve the required compactness that we enjoy today. With the growth of computational power, more functions became available and the calculator design grew. These were gradually clustered above the original square array of $0-9$ digits and basic operators over the decades. However, tradition has a profound weight in design, and the keyboard design changed little over the years, as observed in a more recent (2015) model Casio FX-991ES PLUS 2nd edition. This same trend is also present in other large manufacturers such Texas with their latest Instruments TI-84 Plus CE graphing calculator [7] and Hewlett-Packard with their Prime Graphing Calculator [8]. On advantage of tradition is that users are already accustomed to that particular design, thus facilitating the interface by providing a familiar design with upcoming versions. However, tradition can prevent evolution, especially if the initial design was not necessarily the most efficient to begin with. Indeed, the clustering of 0-9 keys in a matrix array and the most fundamental operators (on the side) is intuitive, but not necessarily time and energy efficient.

Therefore, a gap is seen in how the human mind best interfaces to the design of all those clusters of functions (trigonometric, logarithmic, power) and numbers ($0-9$, decimal place and irrational numbers). How are they to be arranged with respect to each other, to lead to a more intuitive and faster utilization of such a practical tool? The purpose of this article is to study this gap in capability in a rational manner, and present in an argumentative way, a new design that could improve even further the efficiency of their use in our already extensive day-to-day usage. Since improvement in efficiency is systemic, the design change would positively impact in a collateral manner all humankind’s technological processes that involve the usage of a calculator, ranging from classroom education to scientific research and engineering.

It is the hypothesis of this article that it is possible to achieve a more logical arrangement of the numbers, operators and functions keys in a calculator such that typing a calculation becomes more aligned with the natural human thinking process, thus allowing for a more expedite and effortless transformation of thought into computational action, ultimately crystalizing a faster convergence into the desired outcome.

The original keys $0-9$ and fundamental operators of summation, subtraction, multiplication and division are historically laid out in a matrix array design (that is in columns and rows), as shown in Figure 2a. Hence, the process of re-arranging this design follows a logical reasoning that springs from careful observation of the natural human behavior when employing a calculator.

That way, any user can independently verify in practice the soundness of the given arguments. The nature of human beings is to round numbers for simplification or better handling, like $10, 200, 3000, 4000000$, etc, making it a typical choice in many calculations. This suggests that the number zero is used far more in calculations than any other $1-9$ integers. For that reason, in the new design the key zero is located intentionally at the center of the circle, i.e., the center of algebraic calculation activity (Figure 2b). Circular disposition of groups of keys of the same type (the binary operators surrounding the key zero and the integers from $1-9$ surrounding them in turn) allows all keys in a given group to have equal distance to the center of calculation. This is convenient since there is an equal chance of any of the numbers to be selected at any given position of an algebraic operation. The same argument applies to the binary operators. Thus, generally speaking, each circular group is formed by a central button surrounded by an inner and outer track.

Since operators naturally appear in between numbers (as their function is indeed to establish a relation between two quantities), it is only logical to place them in between the numbers $1-9$, and as closely positioned to the center as possible. That way in a natural flow of a calculation, moving inwards and then outwards. That is, after pressing a number combination $1-9$ on the outer track, the focus of the user automatically moves inwards into the inner track to select an operator, and then moves outwards again to select another combination of $1-9$ to complete the overall calculation. That way, similar keys in a group are laid out around in a circle, while different groups are placed in tracks at different radius.

This is not uncommon to the logic behind a DVD player, where the laser head accesses different information by moving to different radial tracks disposed in a circular manner, to perform a given digital operation on a given dataset [9]. This analogy implies that the laser target is the user’s finger or mental focus point, while the laser head movement in circumferential and radial directions is analogous to user’s displacement of his finger in search for the right keys to press. The decimal place key, inherently plays a central role in defining the transition from integers to decimal numbers. Thus, like the key $0$, its location is at the center of the calculation activity. It is differentiated from zero by its smaller size button. An interesting remark is seen when realizing that the usage of multi-layer for storing data in a volumetric format is intuitively achieved with the SHIFT and ALPHA option that, in essence, switch between layer of the keyboard allowing the user to access different layer of information.

Another feature of the circular group design that speeds up calculation is the inclination of the numerical keys (see in Figure 3a). By making all keys in the $1-9$ circular track incline inwards, the action of pressing a key (in the track holding the integers) generates a reactive force with a horizontal component pointing towards the center of the calculation activity (as illustrated in Figure 3b).

This contributes to the spring back of the fingers towards the center, which as discussed before, facilitates the subsequent step to move to another key, whether it be a number or an operator. Moreover, while staring directly at the key zero in Figure 2b, it is possible to see all the numbers and operators using the peripheral vision, which facilitates coordination of every step of the calculation. The importance of this is exemplary illustrated in nature when realizing that many predators often hunt by using their side vision to identify their “targets”.

An analogy to help understand the *modus operandi* of the circular group approach is found in the postal service system. This analogy considers the finger as the parcel, and the house is the key to be pressed. In order for the finger to find the key, it must first have an address destination (which forms a converging path). In courier services, a parcel can travel from any position in the world to another because the world is structured in a logical manner, i.e., continents, countries, cities, districts, streets and house number [10]. A parcel follows a converging path from macro movements (encompassing a large displacement between continents or countries), and then by micro movements (encompassing transit to a particular city and area code) converging to the final street and house. The organization of keys in a calculator should also follow a converging approach for an effective “delivery” of the finger to the required key, which when in chain form operation of a calculation. Therefore, following the postal service approach (which we know to work well worldwide), when the mind is faced with the delivery of a finger to a required key, the best way to organize its path of convergence to its destination is by clustering the calculator keys into circular groups.

In the present case, the three groups are numerical pad and binary operators (Group 1, as displayed in Figure 4), power and logarithmic functions (Group 2) and trigonometric and miscellaneous functions (Group 3). The finger/mind intuitively travels to the center of each circular group first, and then converges to the desired specific key within the group. This converging approach to seek information is seen commonly when searching for an information, comprising of first selecting a volume, then a chapter, a paragraph, and finally reaching the desired sentence. The brace keys were placed on either side of the number plus operators’ pad, following an outer circular track. The rational is that braces typically enclose mathematical operations, often composed only of numbers and operators. Hence, the layout of the keys follows the way a person writes mathematics, making it for a more intuitive alignment between how mathematics is written and how it is typed in a calculator. Keys that command a result, like the equal = and answer ANS functions, were placed on the same outer track, being located in the corners of the pad. The key for all clear AC was placed on the top right corner of this outer track, in a spatial position similar to its predecessor in the classical matrix calculator design.

More specifically, Figure 5 shows the transfer of functions pertaining to the group 2 and 3 from the classical matrix design (Figure 5a) to the circular group design (Figure 5b and 5c). The core of group 2 shown in Figure 5b comprises of the general option $x^{\square}$ placed at the center, surrounded by the inner circle track populated by the most popular variants $x^2$ and $x^3$, the inverse $x^{-1}$, and to the power of a half - i.e., the normal square root $\sqrt{\square}$ . The outer track comprises of the cubic root $\sqrt[3]{\square}$ and the generic root $\sqrt[\square]{\square}$ found at the bottom, and the logarithmic functions (Napierian and based 10 logarithms) positioned readily at the top. Their inverse $10^{\square}$ and $e$ is available via the same keys using the alpha option. Other more general functions such as the fraction and product with fraction, percentage and factorial are also located on the outer track, to the sides. The outer layer of group 3 in Figure 5c comprises of trigonometric functions, with sine and cosine placed at the top, neighbored by tangent and hyperbolic functions on either side. This follows the general left-to-right top-to-bottom approach commonly found in reading and writing. Such tendency is firmly rooted in the human mind, thus to re-use them in a new design will provide by definition more efficiency to the interface.

The key for the constant $π=3.141…$ is placed at the center, with the constant e being also accessed using the alpha option. The bottom half of the outer track comprises of the absolute button and sign change $(-)$, which due to their related functionality in changing the number sign, are placed close together on the left. On the right is a button that provides the golden ratio $φ=1.618…$ that is widely observed in nature and biology in spirals and aesthetic patterns [11, 12], just as $π$ is widely observed (and I used to mathematically describe) circles, hence the golden ratio φ has a key of its own. This is readily followed by the INS or insert button that provides the opposite function of the DEL or delete button at the bottom, thus INS and DEL being next to each other. Moreover, the DEL button is conveniently placed at the bottom, close to the other related key AC or All Clear (located on the right-side track on top). This preserves the DEL-AC pairing already present in the classical matrix design. The inner track contains more calculator governing functions such as recall or RCL, the key ENG to switch numbered displayed into an engineering format, the variants of the memory M function and the S↔D function. Finally, placing these new configurations for group 2 and 3 (Figure 5b and 5c) into the new design on the right of Figure 4, completes the transition from the classical matrix approach in Figure 6a into the circular group approach shown in Figure 6b.

The new circular architecture has the shape of an uneven diamond (in pink in Figure 6b), with each corner representing the center of a group that contains keys of relatable functions. It has $55$ keys – that is $8$ more than the classical array design. In the classical matrix design, the function ON is located in the top right corner (Figure 6a), and the index finger of the user always requires a longer diagonal path to arrive at the center of the keyboard (Figure 6b). In the circular group design, this was moved to the approximate center of the lower triangle of the diamond, or the space between all three circular groups. This shortens the mean distance from the start of a calculation to any of the remaining keys, favoring a more balanced time and energy consumption independent of the initial start choice made. The function OFF, on the other hand, pressing it accidentally in not a desired event, thus it is accessible via the ALPHA option on the top right corner. This final end result of the new circular group logic calculator is illustrated three-dimensional in Figure 7, which was built using the open-source software FreeCAD [13], exported in a tessellated format (.stl), and upon importation, realistically surface-rendered in the open-source software Blender [14].

Keys in a classical calculator keyboard are typically robust and with a good grip, with those in the lower half of the calculator differing slightly in size from those in the upper half. The new calculator design would preserve the aspects of robustness and grip, but the keys would be overall different in size and from each other, except within each track where they are in fact equal. Their curved-shape keys with tight gaps would follow the format found for instance in former generation of mobile phones, like those on the BlackBerry Q10 Smartphone [15]. Since mobile phones are probably more used than calculators, it is only logical to assume that their key pressing reliability is adequate for the working lifecycle of a calculator. The new calculator end-product would probably have keys with a feel and touch in the direction of the former model Casio fxCG20AU, instead of the wider-spaced plain rectangular keys found in the newer model Casio fx-CG50AU.

An attempt is made to quantify how much effort is potentially saved by an average user operating this new calculator design. Assume the bulk effort and time spent in typing a calculation is in translating the index finger back and forth between keys. Here, the distance covered is seen as a logical measurement of effort during a calculation. A comparison is made between the two keyboard designs in a realistic manner. For both cases, the distance between the center of each key was measured. As calculator´s history shows, arithmetic operations were the first to be programmed in a calculator, arguably representing its bulk usage. Functions were introduced later, permitting the execution of more elaborate blocks of operations. Thus, to keep the analysis comprehensive, only the four primary operations of summation, subtraction, multiplication and division are considered. The time between each combination of two possible sequential keys was measured. Around 100 arithmetic calculations were computed, with numbers varying randomly with different length of digits. Based on the time between keys, each calculation was computed and compared. This was done for the classical and new design; in terms of how much overall time the computation would require. Time spent was computed for hundred numbers involving one operation of subtraction. This comprised in changing numbers of digits from $n=1\to5$, where for example $n=1$ gives $x-y$, which is more generically written for any n as $xx…x-yy…y$. Finally, in order to better understand the impact of the choice of operator, the later was changed iteratively (for the same calculations) to all other possibilities. The results from both calculator designs were recorded and compared. All data was computed, post-processed and 3D graph plotted using the open-source software GNU Octave [16]. The random numbers were retrieved from the random number generator inbuilt into Microsoft Excel.

Table 1 shows the distance measured in centimeters in between the numerical keys $0-9$, and from these number to the operators keys $+, -, \times, \div$. This was done for both designs, where the circular group design is placed in the top right triangle, while the classical matrix design is placed on the bottom left triangle. The diagonal of the matrix is populated by an array of zeros, which represent the distance of a key to itself (naturally null).

A back-to-back comparison between the two halves of Table 1 shows that the largest distance in the classical matrix design reduced from $5.55cm$ (between key $0$ and $\div$) to $3.78cm$ in the circular group design (between key $1$ and $6$). And the average distance between two keys reduced from $2.7cm$ to $2.2cm$, respectively, representing a reduction of $20\%$. The end result of the new circular group approach is a more compact keyboard configuration. As an example, Figure 8 shows the key pressing path for the operation $6650-5908$, both for the classical matrix design (Figure 8a) and the circular group design (Figure 8b), showing a lesser chaotic motion of the finger as one transits from the former to the later.

Time is perhaps a more meaningful measure (than distance) of the effort involved in typing any given calculation - that is, the time taken to complete it. For this, a reference is required to link distance with time. In the present study, it is assumed that the index finger covers the distance of $4.26cm$ (the distance between keys $0-9$ in the classical matrix design) in one second. Based on this reference, the distances in Table 1 were converted into time duration in Table 2, respectively.

The diagonal of zeros in Table 2 represents the distance of a key to itself that is zero, making for a converted time of also zero. Another way to look at it is that the time taken to press a key twice is the same for both designs (not dependent on the distance between keys), and thus it is not accounted for in this study. An initial overall way to compare the different designs is by computing how much time that would be required to press all the combination of keys from start-to-finish. The result shows that for the classical matrix keyboard, it would last around 60 seconds, where for the circular design around 50 seconds, which means the user of the new circular group design would finish 10 seconds earlier (with the corresponding lesser effort to do so).

Figure 9a shows a perspective view of a surface map of the distances between two keys converted into time duration for both designs, and Figure 9b shows a corresponding top view. Several differences are observable.

Region $(1)$ shows a smoother transition between keys, inducing a less chaotic thought process of transformation of the key press sequence into actual actionable gestures in the keyboard. The circular group imparts a smoother key transition between keys than the classical matrix format. While the distance between the numbers may have increased slightly in region $(1)$, the insertion of the binary operators within the numbers provides a more reduced travel time from numbers to operators (and back to numbers) as shown in region $(2)$.

In an effort to better understand the impact of the change in calculator design, the time for each specific key-to-key transition from the classical matrix design (upper left triangle half of map in Figure 9b) was subtracted to both designs giving Figure 10. Naturally, the values for the classical matrix design, when subtracted by itself, disappears, while the difference to the circular group design is shown in the lower right diagonal half. Here, the values in red represent an increase in time (with the scale set up to a highest value of $0.5s$), and in blue a reduction (with the scale set up to a lowest value of $-1s$). The end result is not surprising, as if one increases (based on the design change) the distance/time in between the numbers to accommodate the operators within (thus reducing the distance/time of numbers to operators), then it would be expected the inter-number would increase (region 1) in order for the number to operator to reduce (region 2). Figure 10 shows that this transition was achieved successfully, and as it will be shown later, it provides an advantage in speeding up calculations.

In order to better understand how this transition from classical matrix design to circular group design benefits calculation time, a few examples are examined in detail. Figure 11 shows the result of the time count (taken from Table 2) for each of the list of algebraic calculations extracted from several high school exams. It attempts to quantify how much time the students would spend calculating on the pad for both the classical matrix design (CM) and circular group design (CG).

While $4235100$ is simply a long number, the purpose of the example in Figure 11a is to highlight the strength of the classical matrix design that emerges from the close proximity of the numbers (closer than in the circular group design). Simply typing the number takes $+1.5s$ more in the circular group design, however the purpose of a calculator is to perform operations, as that in Figure 11b with the example $1680÷30$. Every strength is accompanied by its weakness, and a shorter distance between numbers means a longer distant from numbers to operators. Thus, the impact of adding one operator results in the time save of $-1.5s$ when switching from the classical matrix to the circular group design. Finally, the example in Figure 11c of $40+100+20+10$ shows that performing multiple calculations in chain with various operators compounds the time saving effect, leading to a greater time saving of $-3.2s$ (than when compared to the single operation in Figure 11b).

The results from a hundred random calculations with varying number of digits is presented in Figure 12, as the difference between the time spent using the circular group design to the classical matrix design. Effectively, twenty computations were done for each number of digits n ranging from 1 on the left, or $x-y$, and increasing up to $5$ on the right, or $xxxxx-yyyyy$. The trend shows the greatest gain appearing in one-digit calculations (i.e., $n=1$) with a computed on average $-1.01s$ time reduction. A considerable gain of the circular group design over the classical matrix design is still present as the number of digits increases to two (i.e., $n=2$) at $-0.86s$. As the number of digits increases to three (i.e., $n=3$), and more time is spent moving in between numbers, some cases show the classical matrix to be faster (due to the greater proximity between numbers than the circular group), however, on average the advantage of the new design is still present with a mean reduction of $-0.50s$. The same trend is observed as the number of digits increases to four (i.e., $n=4$), with some extreme cases being observed for both sides (i.e., extreme 1 favoring the circular group design, and extreme 2 favoring the classical matrix design). These two extremes show that, despite the general trend, the comparison between the two designs can be highly non-linear. This is because of the complicated disposition of the highly complex keyboard designs. To better understand this non-linearity, these two extremes will be further examined below. As the number of digits reaches five (i.e., $n=5$), the mean breaks even, and both designs have equivalent performance.

Figure 13 shows in more detail the time duration spent in calculating the two extremes highlighted in Figure 12. Figure 13a presents extreme 1 (calculation $6650-5908$) that showed an accumulated time gain of $-2.1s$, resulting from the presence of two zeros in the numbers. The key zero is placed at the center in the circular group (CG) design, reducing substantially its overall distance to all other numbers and operators. This produces the recorded large gain in time. Figure 13b presents the other extreme 2 (calculation $3596-9938$) that showed a loss of $+1.4s$. This is a result of a coincidence of the numbers $5,6$ and $9$ being close together in the classical matrix (CM) design and far apart from each other in the circular group design, where the compounded effect of these differences explains for the overall loss.

In order to better understand the impact of choosing other operators, the same $100$ random calculations performed with subtraction in Figure 13 were re-computed for the other possibilities $+,\times, \div$, resulting in Figure 14a-c. While a similar linear trend between Figures 14a-c and Figure 13 is observed, it is important to remember that the two designs also have highly non-linear characteristics.

For comparison, the mean values shown in both Figures 14a-c and Figure 12 were re-plotted in Figure 15. As expected, for low number of digits the circular group design provides faster calculations, starting with operations subtraction and division for one-digit operations (i.e., $n=1$) at $-1.0s$ and gradually increasing with number of digits, until a breakeven point at five digits (i.e., $n=5$). For summation and multiplication, a similar trend is observed but vertically offset, starting at around $-0.4s$ for one-digit operations (i.e., $n=1$), reaching a breakeven point at three digits (i.e., $n=3$), and finally resulting in a penalty for higher number of digits. Overall, three quarters of the calculations favor the circular group design (i.e., three quarters of the lines are located below zero).

Averaging all the means in Figure 15 give an overall average of $-0.26s$ (around one quarter of a second), favoring the circular group design as being overall faster than the classical matrix design. This may not sound like much, but considering a calculation with four successive operations, the compounded time gain would be potentially one full second faster. The reason of the offset distance of the lines for summation/multiplication with respect to those for subtraction/division is easy to understand. Inspecting the classical matrix design in Figure 8a shows that the keys for summation/multiplication are right next to the number’s matrix $1-9$, however, the keys for the subtraction/division are further away to the right. This extra distance of the subtraction/division keys to the number matrix $1-9$ makes for an extra delay in the calculations, which is responsible for the larger difference in calculation time in Figure 15 (i.e., the offset in mean values for calculations using the subtraction/division versus the summation/multiplication operators).

As a practical example, Table 3 shows various fundamental exercises - taken from the Kent State Department of Mathematical Sciences [17] - for which the computing time was evaluated (as done before), for both the classical matrix design and the circular group design calculators. The outcome predicts that using the circular group calculator, the average student could save around 5 seconds (i.e., -18% of the total computing time using the classical calculator) in the execution of these exercises.

While already showing a promising advantage, the benefits reported by the results of this analysis are still conservative, as they do not take into account other aspects inherent to the circular group design that would further speed-up calculation (such as, for example, the impact of a more intuitive design on the speed of converting thought into key-pressing action). One can argue that the evolution of mobile phones has shown interface optimization to be a natural advantage sought out by commercially competitive designs, where more intuitive designs tend to be preferred by users [18]. Another example of the importance of intuitive interfaces is the early evolution of the computer world towards standardization of “plug-and-play” [19]. From an immediate perspective, further research can be done following the same approach involving the various functions’ keys positioned in group 2 (Power and Logarithmic Functions) and group 3 (Trigonometry and Miscellaneous Functions). Here it would be recommended to keep the same distance-to-time conversion reference, to perform the same analysis to determine the distance/time gained involving these keys. From a general perspective, the predicted increase in calculation speed provided by the circular group design could be confirmed in practice by having a group of volunteers conducting a series of calculations back-to-back against the classical matrix design of the casual calculator. This would require the construction of a prototype of the new design, which given the ever increasingly powerful manufacturing techniques like 3D printing, and the existence of the already miniaturized electronics of modern calculators, is an achievable task to any seriously engaged company or research institution.

Design by tradition, while providing safety, may impede evolution. An example is the interface of a calculator, that was built iteratively over the second half of the 20^{th} century, not being challenged or optimized. In this study, it was found that grouping keys in multi-layer circular formats creating focal points of similar functions, provides an interface superior to the classical disposition of keys traditionally organized in a matrix format. Adopting a carefully justified approach, the circular group design was shown to be effective in reducing time and effort (due to its logical intuitive nature) when computing in comparison to the classical matrix format. Moreover, it shown that the time benefit is compounded in longer sequences of operations, yielding ever greater savings. Other foreseen benefits, such as reaction force boosting and thought-to-action acceleration due to a more intuitive layout, whilst not presently quantified, are also discussed in a qualitative manner. Overall, saving seconds with a few operation in a calculator can translate into hours saved at the end of months or even years for both students and professionals. Being systemic, the improvement in such a fundamental technological tool would provide collateral speed-up of all mathematical and scientific processes that rely on a calculator, thus contributing towards the eternal humankind’s endeavor for faster, further and better.

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