# Virtual Reality as an Innovative Tool for Numerosity Perception
## 2 Materials and Methods
### Procedure  
Six participants (five males and one female) took part in the study. The average age of the participants was 34.67 years (range: 28-49 years). All participants had normal or corrected-to-normal vision. The study was approved by the local health service ethics committee (\"Comitato Bioetico dell'Universita di Pisa\", 24 September 2021, n. 31.) and was conducted in accordance with the Declaration of Helsinki.  
Prior to the experiment, a calibration phase was conducted to adjust the HMD for each participant, specifically by setting the interpupillary distance to match their individual needs. This calibration ensured accurate visual alignment and optimal comfort. Participants were then given time to familiarize themselves with the virtual environment, the VR controllers, and the general mechanics of the study.  
The task involved judging two groups of small spheres placed on a virtual table and determining which group contained more spheres. The participant's position, while interacting with the virtual environment, is visually represented by the avatar in Figure 1. It is important to note that this avatar was added only for illustrative purposes in the figure and was not present in the actual virtual environment during the study.  
Participants began the task by standing in front of a starting panel positioned near the virtual table. To start the evaluation, they confirmed their readiness by pressing a virtual button on the panel using the VR controller. Following this confirmation, a series of pairs of red sphere groups appeared on the table. The virtual table measured 0.9 \(\times\) 1.9 m\({}^{2}\) in area and was set at a height of 0.7 m. At the center of the table, a light blue semi-sphere served as a fixation point to help participants maintain their gaze at a consistent location.  
Participants were instructed to stand 0.65 m away from the table's edge (1.1 m from the blue fixation point). Their standing height ranged approximately from 1.70 to 1.85 m, resulting in eye heights between 1.58 m and 1.70 m above the floor. A chair positioned directly behind the participants provided a fixed reference point to standardize positioning during the task.  
The red spheres occupied a circular area with a radius of 18 cm, and the center of this circular area was located 30 cm from the blue fixation point. Each sphere had a diameter of 2 cm. Depending on the experimental condition, the number of spheres within each group varied from 5 to 130. The spatial positions of the spheres were precomputed using MATLAB R2024a (a high-level programming language designed for numerical computation, data analysis, and visualization, widely used in engineering, scientific research, and mathematics), based on a validated algorithm previously employed in similar studies by the research group [35]. These coordinates were then imported into Unity prior to the experiment to ensure precise and reproducible spatial arrangements.  
The VR system dynamically calculated the participant's viewpoint in real-time, rendering the virtual scene based on their physical position. This process ensured that the retinal projection of the spheres remained accurate for each individual observer. As a result, a single sphere subtended a degree of visual angle (dva) of 0.8 degrees, with an average eccentricity of 12.2 dva. The overall spatial extent of the sphere groups spanned approximately 14.5 dva horizontally and 10.6 dva vertically. The dva is a measure of the angle subtended by an object on the retina, determined by its size and distance from the observer.  
During the task, participants were required to determine which group of spheres was more numerous by pressing the trigger button on the corresponding side of the controller. Each pair of sphere groups was presented for 240 ms, but participants were allowed to respond at any time after the initial presentation.