Quadralzahlen: A Rotation-Invariant Numeral System That Reduces Upside-Down Misreads in Real-World Displays

cognizancejournal.com Michél Nguyen, Cognizance Journal of Multidisciplinary Studies, Vol.5, Issue.10, October 2025, pg. 218-230 (An Open Accessible, Multidisciplinary, Fully Refereed and Peer Reviewed Journal) ISSN: 0976-7797 Impact Factor: 5.183 Index Copernicus Value (ICV) = 92.57 ©2025, Cognizance Journal, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden, All Rights Reserved 218 Quadralzahlen: A Rotation-Invariant Numeral System That Reduces Upside-Down Misreads in Real-World Displays Michél Nguyen ORCID: 0000-0001-6834-4422, [email protected], Germany University of the People, Computer Science, USA DOI: 10.47760/cognizance.2025.v05i10.019 Abstract: We introduce Quadralzahlen (QZ), a rotation-invariant numeral system for identifiers and state codes designed to reduce 180° misreads and angle-dependent reading costs in rotationrich contexts (e.g., logistics, healthcare, handheld UIs). QZ combines self-symmetric glyphs (0, 1, 8, X) with pairwise mappings (6–9, 3–E) and typography/OCR guidance. Using ex-ante, insilico evaluations that mirror laboratory recognition, eye-movement proxies, and field picking tasks, plus engineering checks (OCR and display thresholds), we examine whether QZ improves accuracy and time at rotated angles without degrading upright performance after brief familiarisation. Simulations indicate fewer upside-down confusions and flatter angle-time profiles for QZ relative to standard numerals, with OCR remaining near the ceiling at quarterturns without augmentation. We provide design requirements and principles, a dual-notation rollout strategy, and a 2to 4-week micro-pilot protocol for empirical validation. Limitations include reliance on synthetic participants and parametrised penalties; future work should test QZ in live deployments and quantify learning curves and transfer. Keywords: rotation invariance; strobogrammatic; human factors; OCR; logistics; healthcare. 1. Introduction In modern operations like logistics or for handhelds, the readability and accuracy of numerical identifiers are critical to such missions, and these environments are also very rich in rotation – there‘s viewing of labels, displays, and screens from inconsistent angles, inverted orientations, and the like (Brooke, 1996). In these situations, a limitation of Arabic numerals is the difference in orientation. The numbers 6 and 9 or 2 and 5 are often misread when viewed upside down or at bizarre angles. As a result, these readings depend on the angle, which can cause reading errors, delays in verification, and operational safety hazards (Fraleigh, 2003). Even if font and OCR cognizancejournal.com Michél Nguyen, Cognizance Journal of Multidisciplinary Studies, Vol.5, Issue.10, October 2025, pg. 218-230 (An Open Accessible, Multidisciplinary, Fully Refereed and Peer Reviewed Journal) ISSN: 0976-7797 Impact Factor: 5.183 Index Copernicus Value (ICV) = 92.57 ©2025, Cognizance Journal, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden, All Rights Reserved 219 design have become more legible, they are still limited in their strength by the asymmetry of the numeral glyphs. 1.1 Problem Statement Symbols are affected by orientation variance. This means symbols can become confused when rotated. This occurs in logistics labels, clinical coding, and handheld user interface systems. Packages, medical vials, or reading devices may be read from any angle, especially in situations where time is critical or space is limited. Angular deviations, even small ones, can increase human verification loops. Operators must rotate or recheck identifiers to ensure they are correct. OCR engines, which have been trained on upright numerals, may misclassify rotated inputs in digital systems. This will need the model to be retrained or rely on expensive data augmentation pipelines. The issues get much harder as the supply chain gets more complex. Healthcare information is very sensitive. There are many different display formats across devices. 1.2 Objectives and Research Motivation In this paper, we present Quadralzahlen (QZ), a numeral system that is invariant under rotation. The system was designed to deal with the issues arising from the orientation variance. This has been achieved through the use of glyph symmetry and mapping that preserves values. If QZ is designed to achieve value-invariance, one organization can eliminate verification loops and angle-dependent misreads without retraining. QZ helps machines and humans better understand information by providing users with something ‗frictionless‘. The study aims to: 1. Design and instantiate QZ as a functional numeral system combining self-symmetric glyphs (0, 1, 8, X) with pairwise mappings (6–9, 3–E). 2. Formulate design requirements and principles for orientation-robust labeling and display systems that maintain semantic stability under rotation. 3. Evaluate QZ through ex-ante, in-silico simulations, mirroring human recognition, eye-movement proxies, and field-picking tasks to test its efficiency and accuracy at varied rotation angles. 4. Provide preliminary evidence to justify a 2–4-week micro-pilot and subsequent real-world field validation in logistics and healthcare settings. 1.3 Intended Contribution The intended contribution is threefold:  An instantiated (QZ) that embodies the principles of rotation-invariant numeral design.  A series of specifications and principles for orientation-proof typography and labelling systems that can be adapted as conditions change.  Evidence from experiments and simulations shows that QZ can work in practice. It reduces human and machine misreading without the overhead of OCR retraining. The purpose of this study is to extend the work on strobogrammatic numerals, which was mainly theoretical and symmetric in nature, to real displays that exhibit orientation robustness through cognizancejournal.com Michél Nguyen, Cognizance Journal of Multidisciplinary Studies, Vol.5, Issue.10, October 2025, pg. 218-230 (An Open Accessible, Multidisciplinary, Fully Refereed and Peer Reviewed Journal) ISSN: 0976-7797 Impact Factor: 5.183 Index Copernicus Value (ICV) = 92.57 ©2025, Cognizance Journal, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden, All Rights Reserved 220 cognitive ergonomics and computational evaluation. So, that useful contribution goes to humancentred design for information encoding. 1.4. Significance We aim to design and specify a rotation-invariant numeral system (QZ) for identifiers and state codes and to evaluate its expected utility via in-silico experiments (lab, eye-movement proxies, field analogues) and engineering checks (typography/OCR). We test whether QZ reduces upsidedown (180°) misreads and angle-dependent reading time without degrading upright performance after brief familiarisation. 1.5. Aim(s) Rotation and oblique viewing are common in operational settings, yet standard numerals are not robust to these conditions. This paper designs a rotation-invariant numeral system (QZ) and evaluates its performance and deployability. 2. Theoretical Foundations and Related Work The theory underlying Quadralzahlen (QZ) is interdisciplinary in nature and synthesizes theories from the areas of cognitive psychology, design of information systems, typography, and machine perception. QZ does what it does because of cognitive fit. Cognitive fit is defined by Vessey (1991) as the degree of match between the form of information representation and the cognitive requirement of the task. In environments rich in rotation, like logistics or healthcare, conventional numerals impose high mental-rotation costs (Shepard & Metzler, 1971; Hart & Staveland, 1988), as users have to reorient the symbols mentally for recognition. This extra thinking makes doing tasks slower. It also makes it more likely to misread like that. When you change the way symbols look, they still have to be recognisable as the same thing. This is a very moral viewpoint regarding what happens with writing. QZ glyph design embeds rotation invariance to minimize mental transformations, helping to achieve perceptual fluency as glyphs are rotated in arbitrary directions. Looking at design science, QZ‘s drive artefact fits the definitions of Hevner et al. (2004) and Gregor (2006). It highlights the iterative construction and evaluation of innovative artefacts used within a relevant real-world context. QZ is thus positioned as more than just a theory; it is a designed object that embodies a set of design principles for labeling systems that are robust to orientation and can be tested. The UTAUT model (Venkatesh et al., 2003) proposes that user adoption is a function of performance expectancy, effort expectancy, and facilitating conditions within the technological acceptance context. QZ retains recognizable numbers for improved usability without added cognitive or training burden, thus enhancing performance expectancy. Ultimately, QZ allows rotation invariance on the label side of features, whereas machine learning and optimal character recognition systems can achieve rotation invariance on the feature side through data augmentation and group-equivariant architectures (Holmqvist et al., 2011). By cognizancejournal.com Michél Nguyen, Cognizance Journal of Multidisciplinary Studies, Vol.5, Issue.10, October 2025, pg. 218-230 (An Open Accessible, Multidisciplinary, Fully Refereed and Peer Reviewed Journal) ISSN: 0976-7797 Impact Factor: 5.183 Index Copernicus Value (ICV) = 92.57 ©2025, Cognizance Journal, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden, All Rights Reserved 221 relocating this configuration, we reduce the complexity of the hypothesis space for human and machine perception, which hopefully enables robust visual communication with various systems. 3. Design requirements (DR1–DR7) Ten next-generation air defenses will save the world from real attacks because two are limited to land and four are coming from seaports or other transports. These rules freely combine factors and directly arrange analysis on some favorable specifics. 3.1. DR1: Rotation-Robust Legibility We need a letter or symbol where alphabetical order doesn't match the order in quizzes, so it has to be rotated 180 degrees and still be completely clear. Identifiers don't become misleading even when their orientation or position is changed. 3.2. DR2: Upright Performance Parity Once the new numerals become familar, the readability and readability time of the new numerals will be almost identical to standard numerals. Making emulations are safe amid news reading environments. 3.3. DR3: Learnability A question-and-answer website should have intuitive usability, making it easy for someone to learn very quickly. To avoid mistakes in this transition, it makes sense simply to keep writing normal numbers to make learning easier. 3.4. DR4: Typography Constraints The design must adhere to the minimum size, spacing, and type standards to ensure a good look on any surface. These parameters protect our eyes in many different situations. 3.5. DR5: Error Control Encoded identifiers in the QR Decimalization Zone should support error-checking mechanisms, sector handling mechanisms, such as checksum and redundancy checks in harbour channel protocols, to enable double handling of the articulated barge at a terminal that might dispute one of the loadcast facets. 3.6. DR6: OCR Compatibility Optical character recognition systems should maintain a near-perfect accuracy percentage with QZ, without requiring carun rotation techniques. This allows for a smooth process with our current picture recognition system. cognizancejournal.com Michél Nguyen, Cognizance Journal of Multidisciplinary Studies, Vol.5, Issue.10, October 2025, pg. 218-230 (An Open Accessible, Multidisciplinary, Fully Refereed and Peer Reviewed Journal) ISSN: 0976-7797 Impact Factor: 5.183 Index Copernicus Value (ICV) = 92.57 ©2025, Cognizance Journal, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden, All Rights Reserved 222 3.7. DR7: Deployment Practicality Change should consider working with those you already have. To successfully introduce loose notation to a large number of consumers, their role outputs won't fit well into the explanatory group, nor will they ever reestablish a new standard. 4. Artifact instantiation: Quadralzahlen (QZ) QZ comprises a small digit set for identifiers/state codes: 0, 1, 8, X (self-symmetric), and 6–9, 3– E (180° pairs). The radix and grouping marks support composability, while typography guidance sets minimum sizes screen ≥ 12 px; print ≥ 7 pt; LED ≥ 9×13 (preferably 12×16). We recommend error detection via a base-4 Damm variant and short GF (4) codes for corrections in high-stakes contexts (Damm, 2004) or ISO/IEC 7064 check characters. See Figure 1 for the 180° rotation mapping and Figure 2 for the research roadmap. Figure 1: Rotation mapping (180°) for the QZ glyph set. cognizancejournal.com Michél Nguyen, Cognizance Journal of Multidisciplinary Studies, Vol.5, Issue.10, October 2025, pg. 218-230 (An Open Accessible, Multidisciplinary, Fully Refereed and Peer Reviewed Journal) ISSN: 0976-7797 Impact Factor: 5.183 Index Copernicus Value (ICV) = 92.57 ©2025, Cognizance Journal, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden, All Rights Reserved 223 Figure 2: Research roadmap. 5. Design principles (DP1–DP6) Six guiding principles operationalize rotation invariance, cognitive efficiency, and system compatibility in Quadralzahlen-QZ design. These principles are there to connect perceptual design, human–computer interaction, and other designs of encoding information, so that QZ performs robustly against both human and machine. 5.1. DP1: Value-Invariance by Construction QZ naturally achieves rotation invariance through the use of self-symmetric digits (0, 1, 8, X) and pairwise rotational mappings (6–9, 3–E). The design of the glyph of each character ensures it remains the same or predictably changes when turned 180°. This removes any ambiguity at the design level. 5.2. DP2: Dual-Notation Onboarding In the adoption phase, QZ should accompany the normal Arabic numbers flow through, and user confidence should not be affected. The use of two different means of displaying a code helps in gradual familiarization with the code structure. Furthermore, early misreading is avoided, thereby allowing seamless transition in a high-throughput environment. 5.3. DP3: Legibility-First Typography In typography, readability and perceptual clarity must take precedence. This is done by enforcing minimum thresholds for size, contrast, and spacing. Make sure to pay special attention to dense or complex glyphs (for example, 8 and X) to prevent crowding in the character shapes. It can also help maintain legibility between print and screen, including low-resolution LED displays. 5.4. DP4: Human–Machine Alignment The shapes of digits should be designed to be easily interpreted by OCR. The stroke geometry should be consistent, and the contour patterns predictable. Their alignment allows feature learning with less need for data augmentation. This means that an OCR system can recognize digits rotated at arbitrary angles with low error rates and little need for retraining. cognizancejournal.com Michél Nguyen, Cognizance Journal of Multidisciplinary Studies, Vol.5, Issue.10, October 2025, pg. 218-230 (An Open Accessible, Multidisciplinary, Fully Refereed and Peer Reviewed Journal) ISSN: 0976-7797 Impact Factor: 5.183 Index Copernicus Value (ICV) = 92.57 ©2025, Cognizance Journal, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden, All Rights Reserved 224 5.5. DP5: Error-Aware Encoding In situations where operational or safety risks occur due to identification errors, QZ implementation provides support for error detection and localization measures such as collection or check digits, Damm-like algorithms, and GF (4) short codes. These make sure that the data maintains integrity whether entered manually or automatically. 5.6. DP6: Minimal Cognitive Translation The design of the interface and labels should ensure that users do not have to mentally rotate them or remap their cognition as they view the QZ symbols from different angles. This principle highlights the reformulated perspective, where the primary goal is orientation-neutral perception. Thus, observers can effortlessly interpret a given stimulus regardless of the angle of view. 6. Evaluation strategy (ex-ante and ex-post) Quadralzahlen evaluation (QZ) takes place in two phases. This two-phase approach combines a strict validation of Quadralzahlen before deployment (ex-ante) and a real application of Quadralzahlen after deployment (ex-post) on the same system. This combined evaluation system uses computer simulations, human factor modeling, and field metrics to ensure both theoretical soundness and practical feasibility. 6.1. Ex-Ante Evaluation (This Paper) The ex-ante stage consists of a set of in-silico experiments that reproduce the perceptual and operational conditions of lab testing, which makes live testing redundant. These simulations assist in three complementary measurement domains. 1. Laboratory tasks that model symbol recognition in human subjects at various rotation angles provide an estimate of the expected change in accuracy and reaction time (RT) when using QZ numerals instead of conventional (con) numerals. 2. Eye-Movement Proxies: Computational attention models are used for approximating gaze effort and fixation patterns, which are possible indicators of perceptual load and visual search efficiency. 3. The field task proxies — the simulated picking of logistics boxes and labeling of medical products — are used to represent real-world logistics and healthcare workflows, allowing us to evaluate throughput and misreads for various orientation angles. Also, engineering validation checks are performed for testing OCR recognition performance and legibility thresholds on a variety of media (print, LCD, and LED). Overall, the findings provide early insight into the ability of QZ to reduce angle-dependent misreads without impairing upright performance. 6.2. Ex-Post Evaluation (Planned) The ex-post phase will employ dual-notation rollouts for a micro-pilot study of 2–4 weeks in operational settings. This study will include logging rotational exposure to measure actual angle variation and loss of performance under normal viewing conditions. Some of the analytical cognizancejournal.com Michél Nguyen, Cognizance Journal of Multidisciplinary Studies, Vol.5, Issue.10, October 2025, pg. 218-230 (An Open Accessible, Multidisciplinary, Fully Refereed and Peer Reviewed Journal) ISSN: 0976-7797 Impact Factor: 5.183 Index Copernicus Value (ICV) = 92.57 ©2025, Cognizance Journal, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden, All Rights Reserved 225 methods I will use will be mixed-effects regression models and difference-in-differences (DiD) analyses to measure improvements in accuracy, speed and error rates. The System Usability Scale, NASA-TLX workload index and UTAUT-based adoption surveys will be used to collect complementary user feedback. In the end, ROI estimate will be done to access the economic and operational feasibility of QZ for large-scale deployment in logistics and healthcare. 7. Methods (in-silico, ex-ante) Ex-ante analysis of Quadralzahlen (QZ) used a succession of in-silico experiments, which aimed to model the approximation of the laboratory, perceptual, and field conditions under the parameters of the simulation models. The benefit of these virtual experiments is that they enable the systematic study of the effects of rotation, cognitive load, and system performance, and control of the nuisance variables. The simulations were coded in Python and R with reproducible randomization seeds to make them statistically robust. 7.1 Experiment 1: Laboratory Analogue The initial experiment was a simulation of a 2 × 4 factorial design crossing Representation (QZ vs. Standard numerals) and Angle (0, 90, 180, 270). The participants were sampled virtually, N = 40 per cell, and given 40 trials per cell in cell, as is typical with a cognitive recognition paradigm.  The accuracy was created by Bernoulli draws using angle-dependent penalties on standard numerals only, to represent more confusion at reversed orientations.  Reaction time (RT) distributions were modeled using lognormal draws, using a small QZ novelty cost that decreased exponentially with the exposure in trial to represent the effect of learning. This experiment was used to provide an approximation of perceptual accuracy and speed results that would be obtained under a controlled human experiment. 7.2 Experiment 2: Eye-Movement Analogue In order to estimate visual search behaviour, a second simulation simulated an eye-tracking proxy using N = 30 artificial observers. The variables consisted of the duration of fixation (t), number of fixations (f), and the frequency of regression sampled about condition-specific (Representation × Angle) means based on the literature benchmarks. In a mediation analysis, it was determined whether eye-movement proxies mediated the relationship between representation type and simulated reaction time to include the effects of indirect cognitive load. 7.3 Experiment 3: Field-Task Analogue The third experiment replicated real world picking and identification tasks in the case of exposure to rotation. A between-subjects (N = 60) design, simulated 300 picks per participant, of which 3545 percent of the stimuli were rotated. Time per pick and errors per 100 trials were dependent variables, which were determined by exposure-dependent penalties on standard numerals. The model represents logistical and healthcare labeling operational measures. cognizancejournal.com Michél Nguyen, Cognizance Journal of Multidisciplinary Studies, Vol.5, Issue.10, October 2025, pg. 218-230 (An Open Accessible, Multidisciplinary, Fully Refereed and Peer Reviewed Journal) ISSN: 0976-7797 Impact Factor: 5.183 Index Copernicus Value (ICV) = 92.57 ©2025, Cognizance Journal, ZAIN Publications, Fridhemsgatan 62, 112 46, Stockholm, Sweden, All Rights Reserved 226 7.4 Engineering Validation Lastly, engineering tests measured OCR recognition over quarter-turn rotations on various OCR engines and variants of alphabets in two conditions including (i) upright-only training and (ii) rotation-augmented training. Display legibility thresholds were also determined with complementary tests with differences in matrix resolution and pixel density, which ensured the typographic robustness of physical implementation. 8. Results (simulated, ex-ante) In-silico experiments generated some convergent data on the benefits of Quadralzahlen (QZ) in rotation-rich environments as hypothesized. In all simulations, QZ showed consistent performance over viewing angles, decreased perceptual penalties at 180-degree rotation, and fidelity of machine recognition without augmentation (Ware, 2013). Results of each of the simulated experiments are summarized in the following subsections. 8.1 Laboratory Analogue: Representation × Angle Effects Findings in the initial experiment showed that there was a great Representation x Angle interaction, meaning that the standard numerals displayed serious performance loss in the 180° rotation, whereas QZ displayed almost flat accuracy and response-time curve at all angles. There was a sharp decrease in the accuracy of standard numerals inverted, which is corroborated by empirical literature on mental rotation and confusion of symbols. Conversely, QZ had slight short-lived costs in the first trials but soon leveled to equivalent upright performance due to short-term familiarization effects (Wickens et al., 2025). The simulated reaction-time distributions also indicated that QZ needed fewer corrective treatments, which indicated low cognitive load, and mental rotation was not necessary. 8.2 Eye-Movement Analogue: Mediation by Visual Efficiency In the second experiment, the proxies of eye movements established that the fixation durations, counts, and regression frequencies of QZ are lower with rotated angles. Mediation analysis showed that these measures of visual effort mediated to some degree the correlation between representation type and reaction time. Practically, this implies that the rotation-invariant nature of the design of QZ leads to a direct decrease in the visual search and verification cost, which otherwise is involved in the process of the interpretation of inverted symbols. 8.3 Field-Task Analogue: Operational Efficiency The field analogue simulation showed excellent operation advantages of QZ in actual situations. Participants who picked with QZ took less time and fewer rechecks or misreads particularly in conditions where 3545% of the identifiers were rotated. There were increased rotation exposure compounded penalties exhibited by standard numerals, which validated the cumulative cost of orientation variance in workflow throughput. Such results suggest that QZ will be able to enhance accuracy and efficiency in materials movements and healthcare labeling processes without the need to retrain it.