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

Published onMar 08, 2021
New Calculator Design for Efficient Interface based on the Circular Group Approach
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ABSTRACT

Next generation calculators need to be efficient in all its design aspects. Current advanced computation calculators lack an efficient human-machine interface, with its unchanged traditional design traced back to the early 1960’s electro-mechanical commercialized solutions. The present article highlights how the perceived safety of “designing by tradition” can actually prevent evolution, and proposes - in a logical and argumentative way - a new layout based on the circular group approach, which derives from various already successful and working systems operating in society. Numerical comparison of the time spent in computing between the classical matrix calculator layout and the new circular group approach have shown that substantial time-saving can be achieved, in addition to the calculation acceleration effect imparted by aligning the highly organized calculator design with the intuitive nature of the mind of the user. Ultimately, the importance of such an advancement is appreciated when considering the compounding time-saving effect on the millions of users that will spend hours operating calculators in the years to come.

1 Introduction

During the second half of the 20th 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 that time, the initial keyboard design did not change 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 above the yellow box on the right of Figure 1).

Figure 1. Evolution of calculator’s size, price and keyboard design over the second half of the 20th century

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

1.1 Tradition in Design

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 electro-mechanical hybrid to fully electronic – whilst preserving the matrix keyboard 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 over the decades above the rectangular region of the original square array of 090-9 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 Instruments with their 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 in 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 090-9 keys in a matrix array with the most fundamental operators (on the side) is intuitive, but not necessarily time and energy efficient.

1.2 The Research Question

Therefore, a gap is seen in how the human mind best interfaces with the design of all those numbers (090-9, decimal place and irrational numbers) and clusters of functions (trigonometric, logarithmic, power). 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 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 calculators, ranging from classroom education to scientific research and engineering. It is important to early highlight that this is at present an entirely theoretical study, whose purpose is to establish a foundation upon which future experimental dedicated studies can be conducted with participants to generate the necessary data required to analyse the impact of the various diversity factors (e.g., general handling and impact of key size and spacing, user gender, muscle memory, etc). Such studies will determine the practical viability of the new technology within the context of actual human needs to compute, and pave way for further developments of the present theory.

2 Hypothesis

It is the hypothesis of this article that it is possible to achieve a more logical arrangement of 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 in a faster convergence to the desired outcome.

3 Theory

The original keys 090-9 and fundamental operators of summation, subtraction, multiplication and division are historically organized 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 employed when handling a calculator. That way, any user can independently verify in practice the soundness of the given arguments.

Figure 2. Design of the numerical pad with binary operators as a: (a) matrix and (b) circular group

The nature of human beings is to round numbers for simplification or better handling, like 10,200,3000,400000010, 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 191-9 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 (in this case, the binary operators surrounding the key zero and the integers from 191-9 surrounding them in turn) allows all keys in a given group to have equal distance to the center of calculation. Indeed, it is worth noting that the first astrological computer from ancient Greece - the Antikythera mechanism - was circular. This is convenient since there is an equal chance of any of the numbers to be selected at any given position along an algebraic operation. Thus, generally speaking, each circular group is formed by a central button surrounded by an inner and outer track. The present article is the source or origin of this new idea for a calculator circular group design, herein described in detail, and discussed further throughout the following chapters.

3.1 Binary Operators

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 191-9, and as closely as possible positioned to the center. That way a natural flow of calculation, moving inwards and then outwards is established. That is, after pressing a number combination 191-9 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 191-9 to complete the calculation. That way, similar keys are grouped around in a circle, with different groups being placed in tracks at different radius.

3.2 Analogy to DVD Technology

This approach 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 00, its location is at the center of the calculation activity. It is differentiated from zero by its smaller size button. An interesting remark is the realization that the DVD usage of multi-layering for storing data in a volumetric format is already intuitively achieved with the calculator’s SHIFT and ALPHA option that, in essence, switch between layer of the keyboard allowing a user to access different layer of information.

3.3 Force Reactance in Key-Pressing

Another feature of the circular group design that speeds up calculation is the inclination of the numerical keys (seen in Figure 3a). By making all keys in the 191-9 circular track incline inwards, the action of pressing a key (within the track holding the integers) generates a reactive force with a horizontal component pointing towards the center of the calculation activity (as shown 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.

Figure 3. Circular group numerical pad and binary operators: (a) cut perspective view and (b) cut side view

Moreover, while staring directly at the key zero in Figure 3a, it is possible to see all the numbers and operators via the peripheral vision, which facilitates coordination in every step of the calculation. The importance of this is exemplary illustrated in nature many predators that hunt using their side vision to identify “targets”.

3.4 Analogy to Courier Logistics

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 delivering parcel, and the destination house is the key that will be pressed. In order for the finger to find the key, it must first have an address (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 numbers [10]. A parcel follows a converging path from macro movements (encompassing a large displacement between continents or countries), followed by micro movements (encompassing transit to a particular city and area code), finally converging to the desired street and house. Similarly, 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 operations of a calculation. Therefore, following the postal service approach (which we know to work effectively 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 towards its destination is by clustering the keys into circular groups.

3.5 Architecture of the Circular Group Design

In the present case, the three groups to cluster are the numerical pad plus 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 first to the center of each circular group, and then converges to the desired specific key within the group. This converging approach to seek information is seen commonly when searching for literary information, comprising of first selecting a volume, then a chapter, a paragraph, and finally reaching the desired sentence. The parenthesis keys were placed on either side of the number plus operators’ pad, following the disposition of an outer circular track. The rational is that parenthesis typically enclose mathematical operations, often composed only of numbers and operators. Hence, during the calculation the layout of the keys naturally follows the way a person writes mathematics, making it for a more intuitive alignment between how mathematics is traditionally written and how it is typed in a calculator. Keys that command a result, like equal = and answer ANS functions, were placed on the same outer track, 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 that its predecessor - the classical matrix calculator design.

Figure 4. Re-organization of a calculator’s keyboard design based on the circular group approach

3.6 Explanation of Groups 2 and 3

More specifically, Figure 5 shows the transfer of functions pertaining to the groups 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  xx^{\square}  placed at the center, surrounded by the inner circle track populated by the most popular variants x2x^2 and x3x^3, the inverse x1x^{-1}, and the power a half - i.e., the normal square root \sqrt{\square} . The outer track comprises of the cubic root 3\sqrt[3]{\square} and the generic root \sqrt[\square]{\square} at the bottom, and the logarithmic functions (Napierian and based 10 logarithms) positioned readily at the top. Their inverse 1010^{\square} and ee are available via these 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 either side. The outer track of group 3 (Figure 5c) comprises of trigonometric functions, with sine and cosine conveniently placed at the top, neighbored by the tangent and hyperbolic functions on either side. This follows the top-to-bottom approach commonly found in reading and writing. Such tendency is firmly rooted in the human mind, thus to re-use it in a new design provides by definition a more efficient interface.

Figure 5. Arrangement of keys in (a) the classical matrix design, and (b) circular group 2 and (c) circular group 3

In group 3, the key for the universal constant π=3.1415π=3.1415… is placed at the center, with the constant ee being accessed via the same key using the alpha option. The bottom left 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 right is a button that provides the golden ratio φ=1.618φ=1.618… Being widely observed in nature and biology in spirals [11, 12], just as ππ is widely observed and used to describe circles, the golden ratio φ also has a key of its own. This is readily followed by the insert INS button that provides the opposite function of the delete DEL button located at the bottom, thus INS and DEL are next to each other. Moreover, the DEL button is conveniently placed at the bottom, close to the other related key All Clear AC (located on the group 1 outer track on the top right). This preserves the DEL-AC pairing present in its predecessor - the classical matrix design. The inner track contains additional calculator governing functions such as recall RCL, the key ENG to switch displayed number 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 (Figure 6a) into the circular group approach (Figure 6b).

Figure 6. Complete transformation of the calculator design from: (a) the classical matrix to (b) the circular group

3.7 Overview of an End-Product

Figure 6b shows that the new circular architecture has the shape of an uneven diamond (in pink), with each corner of the diamond representing the center of a group that contains keys of relatable functions. It has 5555 keys – that is 88 more than the classical array design. In the classical matrix design, the function ON is located at the top right corner (Figure 6a), making the computing index finger always require an extra longer diagonal path to arrive at the center of the keyboard (Figure 6b), where it starts calculations. In the circular group design, the function ON was moved to the approximate center of the lower triangle of the diamond, i.e. 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 (in terms of time and energy consumption) initial start choice (than with respect to the previous top right corner). 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. The final end result of the new circular group logic calculator is illustrated three-dimensionally 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]. The FreeCAD 3D model file is provided in the open-sharing database Figshare, for the purpose of viewing, modifying and even 3D printing, if desired (download link).

Figure 7. Three-dimensional Computer-Aided Design (CAD) concept of the new calculator based on the circular group approach

3.8 Note on Key Physical Qualities

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 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 curved-liked 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.

4 Analysis

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. The distance in-between the center of each key was measured for both cases. As history shows, arithmetic operations were the first to be programmed in a calculator, arguably representing its bulk usage at the time. Functions were introduced later, permitting the execution of more elaborate blocks of operations. Thus, to keep this analysis comprehensive, only the four primary operations of summation, subtraction, multiplication and division are considered. The distance, and thus time, between each possible combination of two sequential keys was measured. From this, around 100 arithmetic calculations were computed for each operator, with numbers varying randomly with different lengths of digits. Based on the estimated time between keys, the overall time spent in each calculation was computed and compared. This was done both for the classical and new design. The first set being processed was that for the operator of subtraction. This comprised in changing numbers of digits from n=15n=1\to5 in sets of twenty, where for example n=1n=1 gives xyx-y, which is more generically written for any n as xxxyyyxx…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 of summation, product and division. 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 aforementioned random numbers were retrieved from the random number generator inbuilt into Microsoft Excel.

5 Results

5.1 Distance between keys

Table 1 shows the distance measured in centimeters in-between the numerical keys 090-9, and from these number to the operator keys +,,×,÷+, -, \times, \div. This was done for both designs and is presented in Table 1, where the circular group design is placed in the top right triangle, and the classical matrix design is placed on the bottom left triangle. The diagonal of the matrix is naturally populated by an array of zeros, which represent the distance of a key to itself (by definition null).

Table 1. Distance in-between keys for both classical matrix and circular group designs

A back-to-back comparison between the two triangular halves of Table 1 shows that the largest distance in the classical matrix design reduced from 5.55cm5.55cm (between key 00 and ÷\div) to 3.78cm3.78cm in the circular group design (between key 11 and 66). And the averaged distance between two keys reduced from 2.7cm2.7cm to 2.2cm2.2cm, respectively, representing a reduction of 20%20\%. Thus, the end result of the new circular group approach is a more compact keyboard configuration. The example for the operation 665059086650-5908 is displayed in Figure 8, showing the key pressing path, both in the classical matrix design (Figure 8a) and the circular group design (Figure 8b), where the later presents a lesser chaotic motion of the finger.

Figure 8. Example of the calculation 6650-5908 in the: (a) classical matrix design and (b) circular group design

5.2 Time between keys

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 4.26cm4.26cm (the distance between keys 00 and 99 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.

Table 2. Estimated time between keys converted from Table 1 using given reference

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 (i.e., it is not dependent on the distance between keys), and thus it is not accounted for in this study. An initial way to compare the designs is to compute the time 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).

5.3 Comparison between designs

Table 2, comprising of the distances between two keys converted into time duration for both designs, is plotted in a surface map in Figure 9. Figure 9a shows a perspective view, and Figure 9b shows the corresponding top view. Several differences are observable.

Figure 9. Surface graph of time in between keys for both designs: (a) perspective view and (b) top view

Two regions of importance are identified in Figure 9a. Region (1)(1) in the side of the circular group design shows a smoother transition between keys than in the corresponding classical matrix design, which induces a less chaotic thought process in transforming a key-pressing sequence into actual actionable gestures in the keyboard. Region  comprises of the various times between operator keys and numbers, again the ellipses identifying the transitions between classical matrix design and circular group design. Overall, while the distance between the numbers may have increased slightly in the transition between regions (1)(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 the transitions between regions (2)(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 between both designs giving Figure 10. Naturally, the values for the classical matrix design, when subtracted by themselves, disappear, while their 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 to 0.5s0.5s), and in blue a reduction (with the scale lowest value set up to 1s-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 that the inter-number distance/time would also increase (region 1) in order for the number to operator distance/time to reduce (region 2). Figure 10 shows that this favoring a lower number to operator distance/time was achieved successfully, and as it will be shown later, it provides an advantage in speeding up calculations.

Figure 10. Surface graph of time difference between the two designs

5.4 Examples

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 (computed using Table 2) for several algebraic calculations extracted from high school exams. It attempts to quantify how much time the students would spend for both the classical matrix design (CM) and circular group design (CG).

Figure 11. Time spent in executing calculations: (a) 4235100, (b) 1680÷30 and (c) 40+100+20+10

While 42351004235100 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+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÷301680÷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-1.5s when switching from the classical matrix to the circular group design. Finally, the example in Figure 11c of 40+100+20+1040+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-3.2s (than in comparison to the former single operation example shown in Figure 11b).

5.5 Hundred Calculations with Minus

The results from a hundred random calculations with varying number of digits is presented in Figure 12, showing the difference between the time spent using the circular group design and the classical matrix design. Effectively, twenty computations were done for each number of digits n ranging from 1 or xyx-y (on the left of the graph), and increasing up to 55 or xxxxxyyyyyxxxxx-yyyyy (on the right). The trend shows the greatest gain appearing in one-digit calculations (i.e., n=1n=1) with an averaged computed time reduction of 1.01s-1.01s. 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=2n=2) with 0.86s-0.86s. As the number of digits increases to three (i.e., n=3n=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-0.50s. The same trend is observed as the number of digits increases to four (i.e., n=4n=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=5n=5), the mean breaks even, and both designs have equivalent performance.

Figure 12. Time difference of several subtractions between two random integers with increasing digits

Figure 13 shows in more detail the time duration spent in calculating the two extremes highlighted in Figure 12. Figure 13a presents extreme 1 (i.e., calculation 665059086650-5908) that showed an accumulated time gain of 2.1s-2.1s, resulting from the presence of two zeros in the numbers. This occurs because the key zero is placed at the center of 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 (i.e., calculation 359699383596-9938) that showed a loss of +1.4s+1.4s. This is a result of a coincidence of the numbers 55, 66 and 99 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 the overall loss.

Figure 13. Time spent in executing the extreme calculations: (a) 6650-5908 and (b) 3596-9938

5.6 Hundred Calculations with Other Operators

In order to better understand the impact of choosing other operators, the same 100100 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.

Figure 14. Impact of operator choice on the time difference in calculation between the two designs: (a) summation, (b) multiplication and (c) division

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=1n=1) at 1.0s-1.0s and gradually increasing with number of digits, until a breakeven point at five digits (i.e., n=5n=5). For summation and multiplication, a similar trend is observed but vertically offset, starting at around 0.4s-0.4s for one-digit operations (i.e., n=1n=1), reaching a breakeven point at three digits (i.e., n=3n=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).

5.7 Overall Observation

Averaging all the means in Figure 15 give an overall average of 0.26s-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 191-9, 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 191-9 makes for an extra delay in the calculations, which is responsible for the offset in Figure 15 (i.e., the difference in mean values for calculations using the subtraction/division versus the summation/multiplication operators).

Figure 15. Mean time difference between designs, for all operators +,-,×,÷

5.8 University Example

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 by using the circular group calculator, the average student could save around 5 seconds (i.e., -18% of the total computing time) in the execution of these exercises.

Table 3. Comparison between several textbook exercises done using the two calculator designs

Being a purely theoretical concept at the moment, there is unfortunately no additional user data to share beyond that generated by the present author to highlight the potential usefulness of the new design. Further practical assessments will undoubtedly require the involvement of a college or university in order to conduct a field study, where the collected data from student’s – in terms of preferences and wishes - can be feedback to the design process as further improvements.

There are various math exams students take yearly that require the usage of calculators. In the UK, these are GCSE, A-Level, Nat.5, Higher, Junior, Leaving, Diploma and C&G [18], while in the US are SAT, AP, PSAT/NMSQT, ACT and IB [19]. According to the US National Centre for Education Statistics [20], in the fall 2017, some 5.7 million students were enrolled in private elementary and secondary schools. Since mathematics is a foundation subject, the usage of calculators in the exams of this body of students becomes a business opportunity. The global business of producing calculators implies that large manufacturers like Casio, Hewlett-Packard, Texas Instruments, etc need to be informed of what policies apply to calculator usage in exams, and which models are acceptable in which exams. According to Casio (Table 4), their model line fx-83GTX – of which the present study uses as a reference the particular case Casio fx-83GT PLUS (issued in 2010) - is permitted in all UK exams.

Table 4. All UK, Irish and IB Diploma examinations in which Casio models fx-83GTX are permitted

Since the new proposed circular group calculator employs the same functions and capability as the fx-83GT PLUS, it is anticipated that it will be also equally permitted in said exams. According to the present study, the benefit of nearly 20% time reduction in calculations expected would favor the usage of the new circular group design in those millions of school exams that occur on a yearly basis. Exact estimation of the time gain would require a dedicate avenue of research, preferably encompassing an eager body of education like a volunteer secondary school.

6 Future Work

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 [21]. Another example of the importance of intuitive interfaces is the early evolution of the computer world towards standardization of “plug-and-play” [22]. From an immediate perspective, further research can be done following the same approach (used here for group 1) 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 (as defined earlier), to perform the same analysis to determine the distance/time gained involving these other keys in groups 2 and 3. From a general perspective, the predicted increase in calculation speed when using 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 reachable by any seriously engaged company or research institution. Alternatively, the creation of a phone/computer app of the new circular group calculator is a relatively more inexpensive and less time/effort consuming way forward, that any interested programmer can undertake. Here, the task of developing new hardware (including the necessary cost estimate in building a prototype) is overall more demanding than programming software. This could be a step-in-between the two, where the support shown by users of the app version could contribute to building a case with investors wishing to finance the building of a prototype, in view of a potential future commercialization.

7 Conclusion

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 20th 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 both time and effort (due to its logical intuitive nature) when computing in comparison to the classical matrix format. Moreover, 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 operations 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.

8 References

[1] n.d. (2021)“The Old Calculator Web Forum”. Retrieved from https://www.oldcalculatormuseum.com

[2] n.d. (2021) “Vintage Calculator Web Museum”. Retrieved from http://www.vintagecalculators.com

[3] n.d. (2020) “The Calculator Homepage”. Retrieved from https://www.calculator.org

[4] Valéry, N. (1975). “The electronic slide-rule comes of age”. New Scientist 65(938) pp. 506-511

[5] n.d. (2010). “fx-83GT PLUS User’s Guide”. Retrieved from https://support.casio.com/storage/en/manual/pdf/EN/004/fx-83_85GT_PLUS_EN.pdf

[6] n.d. (2017). “fx-CG50 AU Software User’s Guide”. pp. 541-559 Retrieved from https://support.casio.com/storage/en/manual/pdf/EN/004/fx-CG50AU_Soft_v350_EN.pdf

[7] n.d. (2020). “Getting Started with the TI-84 Plus CE Graphing Calculator”. p. 4 Retrieved from https://education.ti.com/download/en/ed-tech/3BBF042421644CE2AF713484B03A8B11/D82D7288C1904A5486B1372052644810/TI-84_Plus_CE_EN.pdf

[8] n.d. (2015). “Datasheet: HP Prime Graphing calculator”. p. 1 Retrieved from http://www.hp.com/united-states/HP-Prime-Graphing-Calculator-Datashet.pdf

[9] Burr, G. W. (2003). Three-Dimensional Optical Storage (PDF). SPIE Conference on Nano-and Micro-Optics for Information Systems. pp. 5225–16.

[10] n.d. (2014). “Parcel Labeling Guide”. United States Postal Service - New Products and Innovations. Retrieved from https://postalpro.usps.com/parcellabelingguide

[11] Marshall, J. (2012). “How to Use the Golden Ratio to Take Better Pictures”. Retrieved from https://www.scientificamerican.com/article/how-to-use-the-golden-ratio-in-photography/

[12] Cook, T. A. (1914). The Curves of Life. London: Constable and Company Ltd. pp. 390-400.

[13] van Havre, Y. et al. (2021). “FreeCAD - A Manual”. Retrieved from https://freecadweb.org/manual/a-freecad-manual.pdf

[14] Blain, J. M. (2012). “An Introduction to Blender 3D: A Book for Beginners”. Retrieved from https://download.blender.org/documentation/pdf/

[15] n.d. (2016). “Userguide: BlackBerry Q10 Smartphone”. v10.3.3 p. 6 Retrieved from https://docs.blackberry.com/content/dam/docs-blackberry-com/release-pdfs/en/device-user-guides/BlackBerry-Q10-Smartphone-10.3.3-User-Guide-en.pdf

[16] Eaton, J. W., Bateman, D., Hauberg, S. and Wehbring, R. (2021). “GNU Octave: A high-level interactive language for numerical computations”. Retrieved from https://octave.org/octave.pdf

[17] n.d. (2010). MATH 10021: Core Mathematics I. Kent State Department of Mathematical Sciences. pp. 7-28. Retrieved from http://www.math.kent.edu/ebooks/10021/CMI.pdf

[19] n.d. (2021). “Texas Instruments - Exams Resources”. Retrieved from https://education.ti.com/en/resources/test-preparation

[20] Hussar, B., NCES; Zhang, J., Hein, S., Wang, K., Roberts, A., Cui, J., Smith, M., AIR; Bullock Mann, F., Barmer, A., and Dilig, R., RTI (2020). “The Condition of Education 2020“. Institute of Education Sciences / National Center for Education Statistics 2020-144 U.S. Department of Education. Retrieved from https://nces.ed.gov/pubs2020/2020144.pdf

[21] Myers, B.A, Nichols, J., Wobbrock, J. O., and Miller, R. C. (2004). “Taking handheld devices to the next level”. IEEE Computer Journal 37(12) pp. 36−43.

[22] Laing, G. (2004). “Digital Retro: The Evolution and Design of the Personal Computer”. Publisher: Sybex; New title edition















Comments
6
Luis Teia: A promotional video has been created, and is available to view and download at the free database platform Figshare, accessible via the link below: https://figshare.com/articles/media/_VIDEO_New_Circular_Group_Calculator_-_Promotion_TJOE_Luis_Teia_2021_/14766411
Devin Berg: Dear author, I am please to accept your submission for inclusion in The Journal of Open Engineering. Thank you for your participation in our peer review process. Regards, Devin Berg Editor
AG
Alicia Gibb: Thank you Luis! All my concerns have been addressed. I would approve this paper.
Luis Teia: Dear Cheng Liu, Thank you for your interest in the article, for taking the time and effort to review it, and for all your valuable comments. Answers were formulated for each, which are kindly presented below for each corresponding section. I hope the new modified draft (to be released by the editor) meets your expectations with satisfaction. Best Regards, Luis Teia ———————————————————————————————————————————— TECHNICAL SOUNDNESS Thank you for the positive appreciation. CLARITY The following paragraph has been modified as per reviewer recommendation. “This same trend is also present in other large manufacturers such as Texas Instruments with their latest TI-84 Plus CE graphing calculator [7] and Hewlett-Packard with their Prime Graphing Calculator [8].” Concerning section 5.1, Distance between keys. Indeed, a trade-off study involving compactness and usability is important, and is surely to be considered/discussed in a future work. This will involve conducting a dedicated session with volunteers to generate the necessary data required to analyse the impact of the various diversity factors (including finger size, and thumbs versus index fingers) on how users operate the new calculator versus the existing design. The following sentence has been included (at the end of section 1.2, The Research Question), specifically for this purpose. “It is important to early highlight that this is at present an entirely theoretical study, whose purpose is to establish a foundation upon which future experimental dedicated studies can be conducted with participants to generate the necessary data required to analyse the impact of the various diversity factors (e.g., general handling and impact of key size and spacing, user gender, muscle memory, etc). Such studies will determine the practical viability of the new technology within the context of actual human needs to compute, and pave way for further developments of the present theory.” COMPLETENESS Concerning section 5.8, University Example. Being a purely theoretical concept at the moment, there is unfortunately no additional user data to share at present, beyond that generated by the present author to highlight the potential usefulness of the new design. Further practical assessments will undoubtedly require the involvement of a college or university in order to conduct a field study, where the collected data from student’s – in terms of preferences and wishes - can be feedback to the design process as additional improvements. The need to conduct this field study has been added in the following sentence to the body of text within this section. “Being a purely theoretical concept at the moment, there is unfortunately no additional user data to share beyond that generated by the present author to highlight the potential usefulness of the new design. Further practical assessments will undoubtedly require the involvement of a college or university in order to conduct a field study, where the collected data from student’s – in terms of preferences and wishes - can be feedback to the design process as further improvements.” OPENNESS AND REPRODUCIBILITY The phone/computer app is a great idea (that any interested programmer can undertake), being a better (i.e. inexpensive and less time/effort consuming) immediate option than pursuing the construction of a working prototype – a task that would involve developing new hardware, overall being more demanding than software programming. A sentence has been added to the text (at the end of chapter 6, Future Work) highlighting this phone/computer app idea as a helpful alternative option. Indeed, the support shown by users with the app version could contribute to building a case with investors wishing to finance the building of a prototype, in view of potential future commercialization (following in the footsteps of Texas Instruments, Hewlett-Packard and Casio). Therefore, a supplementary sentence has been added highlighting the need for a cost assessment during the future creation of a new prototype, as part of an approach for marketing. “Alternatively, the creation of a phone/computer app of the new circular group calculator is a relatively more inexpensive and less time/effort consuming way forward, that any interested programmer can undertake. Here, the task of developing new hardware (including the necessary cost estimate in building a prototype) is overall more demanding than programming software. This could be a step-in-between the two, where the support shown by users of the app version could contribute to building a case with investors wishing to finance the building of a prototype, in view of a potential future commercialization.”
CL
Cheng Liu: Thanks for inviting me as a reviewer. The paper sparks a new idea to create an intuitive and efficient calculator.   TECHNICAL SOUNDNESS The technical contents are sound for a theoretical study of a new calculator design. The graphs, charts, and figures are clear for their purposes. CLARITY In section 1.1, for consistency with the use of the company names, it is recommended to revise it to: “This same trend is also present in other large manufacturers such as Texas Instruments with their TI-84 Plus CE graphing calculator [7] and Hewlett-Packard with their Prime Graphing Calculator [8].” In section 5.1, the data in the paper shows that the circular group design is more compact keyboard configuration than the classic style. It is a tradeoff between compactness and usability, especially for people with large fingers. A further study involving the compact circular key design for users with finger size variations, genders, can be considered. COMPLETENESS In the section 5.8, University Example. it may be worth of including some user experience data for the new calculator with a circular group design. A survey related to student’s preferences or their use cases can be included as well. OPENNESS AND REPRODUCIBILITY The new calculator with a circular group design can be created as a phone/computer app. If a physical prototype is considered, it may be worth of looking at economic justifications to manufacture a prototype. This typically involves with a new printed circuit board to fit the shapes and positions of the rotary keys on the calculator. In that case, a cost estimate/comparison with the classic calculators can be a useful information to attract any seriously engaged company or research institution to a new calculator.
AG
Alicia Gibb: Thanks for including me as a reviewer, it was all very interesting to read this, I love out of the box thinking for design! TECHNICAL SOUNDNESS The technical aspects are sound as far as the theory goes. I am wondering if you will end up with different results with a study involving humans because of muscle memory, finger size variations with regard to gender, or right / left handedness. I would be more transparent earlier in the paper that the data is theoretical and not derived from a study with participants. I could see that fitting into section 1.2 or in the introduction of section 3. You may also offer an explanation of why this is theoretical for now, is it limitations on producing the hardware or COVID related, etc.? CLARITY The writing and graphs / charts were very clear. The only typo I saw was on page 15, there is a Region missing it's number. There needs to be some clarity around the time specified to reach buttons. Where your averages come from for time of the average reach between numbers would be good to clarify. Do you have a reference that cites a particular time to distance or was it an experiment you did with your own time to reach a button?    COMPLETENESS I see two areas to improve on completeness. I am wondering if there is research to reference regarding dials on interfaces in other technology such as the rotary phone or ipods, and why designers moved away from numbers in a circle for more modern phones; as well as how cell phones in general have changed our behavior of typing with fingers to typing with thumbs, which the calculator would benefit from that learned behavior.  Is there a particular ratio you used with the layout of the calculator keyboard, is the Golden Ratio used in the design itself? Was there any research done to the comfort of how far thumbs can reach before repositioning the hand, that could potentially take more time. In the section discussing high school exams and time saving, it may be worth stating how many high schools require calculators for math classes (or at least the fact that that is still a viable industry), how many exams a student takes in a year, and how many calculations are on the average exam and therefore time saved per exam. Some references would be good in the example section as well if you are including this data.  Finally, this might be too far out in left field, so take it with a grain of salt, in the Intro when discussing design, it may be worth noting the first computer from ancient Greece was circular.  OPENNESS AND REPRODUCIBILITY I would expect to see a section on Source - where the source lives currently, what licenses are being used and any supporting materials (BOMs, instructions, etc). I did not see any links in the paper relating to source and assume these would need to be added to be published.
Luis Teia: Dear Alicia, I have addressed all the points raised, and wrote below the proposed answers. Hoping that you find them satisfactory, if there is any aspect that remains unclear, please let me know and I will gladly further expand them accordingly. The corrections defined below will become visible/available when the modified draft is releaed. Kind Regards, Luis Teia  —————————————————————————————————————————— TECHNICAL SOUNDNESS The technical aspects are sound as far as the theory goes. I am wondering if you will end up with different results with a study involving humans because of muscle memory, finger size variations with regard to gender, or right / left handedness. I would be more transparent earlier in the paper that the data is theoretical and not derived from a study with participants. I could see that fitting into section 1.2 or in the introduction of section 3. You may also offer an explanation of why this is theoretical for now, is it limitations on producing the hardware or COVID related, etc.? The sentence below has been included (at the end of section 1.2, The Research Question) to early clarify that at this point this is an entirely theoretical study, laying the foundation for a potential future test involving volunteers for correlation. At this point in the overall project, the aim is to establish the theory so that subsequently, with appropriate funding and support, hardware/protype of such calculator can be built and used in a testing campaign involving human subjects (which is somewhat more challenging, considering the current impact of the global health situation on social distancing). “It is important to early highlight that this is at present an entirely theoretical study, whose purpose is to establish a foundation upon which future experimental dedicated studies can be conducted with participants to generate the necessary data required to analyse the impact of the various diversity factors (e.g., general handling and impact of key size and spacing, user gender, muscle memory, etc). Such studies will determine the practical viability of the new technology within the context of actual human needs to compute, and pave way for further developments of the present theory.”  CLARITY The writing and graphs / charts were very clear. The only typo I saw was on page 15, there is a Region missing it's number. Typo was corrected, thank you. There needs to be some clarity around the time specified to reach buttons. Where your averages come from for time of the average reach between numbers would be good to clarify. Do you have a reference that cites a particular time to distance or was it an experiment you did with your own time to reach a button? To my knowledge such time estimates are non-existent, thus I had to take my own example (and consider what it would be a sensible average value) to formulate the connection between distance between keys and time spent transiting between keys. In this case, a few calculations involving pressing 0 followed by 9 were conducted, and the time measured between these two keys was on average a second. I believe this is a reference much inline with our own life and biological rhythm, as for example the time between two heart beats while at rest generally rounds to a second. The second is a unit measure of time of which other durations are measured against/multiples, facilitating the comparison by using a more intuitive time reference. The important aspect to remember here is that even thought the time reference can change from person to person, the percentage gain obtained by changing from the classical matrix format to the circular group format remains the same, as it is independent of the chosen time reference (as long as the same reference is applied commonly to both concepts). That is, the percentage gain is defined by the physical distance between the keys. COMPLETENESS I see two areas to improve on completeness. I am wondering if there is research to reference regarding dials on interfaces in other technology such as the rotary phone or ipods, and why designers moved away from numbers in a circle for more modern phones; as well as how cell phones in general have changed our behavior of typing with fingers to typing with thumbs, which the calculator would benefit from that learned behavior.  I agree, this is a complex problem with multiple influencing variables and factors. While handling a calculator may be subjective, and can vary from person to person, the distance between keys is defined by design, and is only dependent on the device (i.e., it is a physical measure). If any comparison is attempted, this is a starting point for a common ground comparison. A forty-year-old accountant may be inclined to use the traditional one hand typing with the calculator resting on his desk while he writes on a computer, while a fifteen-year-old student may prefer to use it with both hands and thumbs as he uses his mobile phone. The answer to this is beyond the scope of this theoretical paper, as it implies opening or dealing with several avenues of research. Regarding the evolution of phones, the following can be said – the keyboard of mobile phones, in its QWERTY letter layout present a stagnation in evolution even greater than calculators. The letter layout, like the commonly known QWERTY arrangement, of mobile phones and computer keyboards alike remained the same since the beginning of the 19th century. This will be further discussed as part of a follow-on article on the evolution of keyboards that the present author is currently writing. Regarding the comparison between using thumbs or the conventional index finger, a comparison between the two to determine which is more convenient to which personality and occupation is beyond the scope of this research, however at this point there no information that discourages one to think that in a back-to-back comparison between the classical matrix design versus the new circular design, an improvement would be noticed in any case simply because while the typing fingers may change, the single focus of the mind as it transits from one key to the next remains the same (being using an index finger or thumbs), making thus for time gained as one’s mind transits between keys, one-by-one at the time, just as studied in the present article. Is there a particular ratio you used with the layout of the calculator keyboard, is the Golden Ratio used in the design itself? Was there any research done to the comfort of how far thumbs can reach before repositioning the hand, that could potentially take more time. For back-to-back comparison, the general outer dimensions of the calculator were kept the same as that of Casio fx-83 GT PLUS. The keys were made as large as possible to make full use of the available space. Following on the footsteps of Leonardo da Vinci, you are right in remarking that the Golden Ratio would provide a better option for an ideal height-width ratio of a calculator. Since both calculators had the same general dimension, it is plausible to think that if a given user can reach all keys with his/her thumbs in the classical design (an exercise I have done personally holding it with both hands, and found that any key can be reached by at least one thumb), then the new calculator would most likely provide the same reach as it has the same size. A further study could be conducted to cover this particular topic, of which a part would emerge undoubtedly from the field study with volunteers. In the section discussing high school exams and time saving, it may be worth stating how many high schools require calculators for math classes (or at least the fact that that is still a viable industry), how many exams a student takes in a year, and how many calculations are on the average exam and therefore time saved per exam. Some references would be good in the example section as well if you are including this data.  I will dig out this information, to the best of my abilities. This is a support analysis; hence I hope to provide a sufficient (not exhaustive) amount of data that can satisfactorily answer this query. Finally, this might be too far out in left field, so take it with a grain of salt, in the Intro when discussing design, it may be worth noting the first computer from ancient Greece was circular. This is interesting, I did not know that. I see it as a positive thing, indeed the present article could be regarded as returning to the origins of computing in a circular manner, as the Antikythera mechanism in so does. I have introduced the following sentence (in the introduction of chapter 3, Theory) to acknowledge this fact. “Indeed, it is worth noting that the first astrological computer from ancient Greece - the Antikythera mechanism - was circular.” OPENNESS AND REPRODUCIBILITY I would expect to see a section on Source - where the source lives currently, what licenses are being used and any supporting materials (BOMs, instructions, etc). I did not see any links in the paper relating to source and assume these would need to be added to be published. I am not accustomed to the term Source, but I imagine it is where the design comes from. Please correct me if I am miss interpreting. I designed this calculator from scratch, and this article is the first time ever such a design is shown publicly. So, I would say this article is the source of the design, created using as tools the opensource cited software. A sentence has been added to acknowledge this. “The present article is the source or origin of this new idea for a calculator circular group design, herein described in detail, and discussed further throughout the following chapters.”  
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