The number system of Solaria is a positional system in base 7, graphically grounded in the numbers 1, 2 and 5. Its mathematical structure belongs to the standard class of positional numeral systems: each digit assumes value based on the position it occupies, and the transition to the next order occurs after 6. The number seven, therefore, has no autonomous digit, but is written 10.
The distinctive feature of the system lies not in its arithmetic functioning, but in the shape of the glyphs, their symbolic meaning, and the relationship between the cycle of seven and the internal subdivision of the unit. The operative digits are seven: 0, 1, 2, 3, 4, 5, and 6. The system may be called the heptimal system, that is, founded on seven.
Digits
The principal glyphs of the system are 2 and 5. The 5 is represented by a form resembling an S, while the 2 is represented by a reversed S. The two signs are therefore mirror images of each other and constitute the generative pair of the system.
The 0 is represented by a form resembling a vertical 8. This sign arises from the graphic union of 2 and 5: the two mirror-image S shapes, joined together, produce a double figure reminiscent of infinity. For this reason, the zero of Solaria is understood not merely as the absence of value, but also as a sign of union, recomposition, and completion.
The other digits derive from 2 and 5 through a small stroke above or below. A lower stroke indicates a decrease of one unit; an upper stroke indicates an increase of one unit. The number 1 therefore derives from 2 diminished by 1, 3 from 2 increased by 1, 4 from 5 diminished by 1, and 6 from 5 increased by 1.
| Value | Glyph description | Symbolic origin |
|---|---|---|
| 0 | vertical 8 shape | union of 2 and 5 |
| 1 | shape of 2 with lower stroke | 2 − 1 |
| 2 | reversed S shape | mirror glyph |
| 3 | shape of 2 with upper stroke | 2 + 1 |
| 4 | shape of 5 with lower stroke | 5 − 1 |
| 5 | S shape | primary glyph |
| 6 | shape of 5 with upper stroke | 5 + 1 |
The strokes appearing in the glyphs are not operative signs, but parts of the digit. Once the glyph is established, it is read directly as an autonomous digit. The number 3, for example, need not be recalculated every time as 2 + 1: this relationship explains its graphic origin, not its everyday use.
Base 7
The system uses base 7. This means that, after 6, the digit cycle closes and advances to the next position. The initial sequence is therefore:
0, 1, 2, 3, 4, 5, 6, 10, 11, 12, 13, 14, 15, 16, 20
In this notation, 10 equals seven in decimal, 100 equals forty-nine in decimal, 1000 equals three hundred and forty-three in decimal, and so on.
Seven is therefore not a visible digit, but the principle governing the transition from one order to the next. This characteristic also carries symbolic value: seven represents the completion of the cycle, not an element internal to the cycle itself.
The Relationship between 2 and 5
The relationship between 2 and 5 is the symbolic center of the system. The two glyphs are mirror images: 5 is represented by an S, while 2 is represented by a reversed S. When they are joined graphically, the two signs generate a form resembling a vertical 8, which in the Solarian system corresponds to zero.
The same relationship is found on the arithmetic plane. In base 7, 2 + 5 completes the digit cycle and produces 10. This means that the union of 2 and 5 generates a new unit in the higher order and leaves zero in the lower order.
For this reason the relationship between 2 and 5 is particularly important: what graphically produces zero, arithmetically produces the passage through zero. The two signs, taken separately, represent a mirror-image pair; united, they indicate the completion of the cycle and the beginning of a new order.
Subdivision of the Unit
Alongside the positional structure in base 7, the Solarian number system provides for an internal subdivision of the unit into twelve parts. This subdivision does not modify the base of the numeral system, but concerns the way in which a unit can be measured, articulated, or divided.
It is therefore necessary to distinguish between pure number and measure. In pure number, even the fractional part can be treated according to the positional logic of base 7. In measure, however, what precedes the separator indicates complete units, while what follows the separator indicates the internal parts of the unit.
The principle is similar to that of minutes and seconds in the ordinary system. When writing a duration, numbers are expressed with common digits, but the carry does not always occur at ten: 60 seconds form 1 minute, 60 minutes form 1 hour. The number-writing system remains the usual one, but the measure follows its own internal articulation.
In the Solarian system, something analogous occurs. Units are counted in base 7, but each unit can be divided into twelve internal parts. When the internal parts reach the value twelve, they form a new unit.
Since twelve in decimal is written 15 in the heptimal system, one unit contains 15 internal parts in Solarian notation. Consequently, the transition from an internal part to the unit occurs upon reaching 15, not because the system ceases to be in base 7, but because the unit is articulated into twelve parts. In essence, one unit consists of 7 (10) + 5.
Measure Notation
To avoid ambiguity, measure notation must be distinguished from standard fractional notation. The comma used in a mathematical sense indicates a positional fraction in base 7. Solarian measure, on the other hand, uses a different separator, to indicate that the subsequent part does not represent sevenths, forty-ninths, or negative powers of 7, but twelfths of the unit.
The separator used in measure is a central dot. A notation such as:
3·5
is therefore read as three units and five internal parts of the unit. In modern terms, it means three units and five twelfths.
When instead one writes:
3,5
in a purely positional mathematical notation, the meaning is different: three units and five sevenths.
The distinction between the two separators is important. Positional notation belongs to general numerical calculation; measure notation belongs to quantities articulated in units and internal parts.
In the Solarian system, internal parts can still be counted using the same heptimal numbering (just as 1.5 ordinary minutes is 90 seconds). The value twelve is written 15. Therefore the last internal part before the return to the unit corresponds to 14, that is, eleven internal parts; upon reaching 15 internal parts, one unit is completed.
The Relationship between 7 and 12
The Solarian number system is founded on base 7, but its symbology connects seven to twelve. Seven represents the complete cycle of positions, while twelve represents the internal subdivision of each unit of the cycle.
In decimal notation, this relationship can be expressed as:
7 × 12 = 84
In heptimal notation, the same relationship becomes:
10 × 15 = 150
This second form is particularly important because it shows the relationship from within the system itself. Seven is written 10, twelve is written 15, and their product is written 150.
For this reason the number 84 holds special value. It does not replace the number 7 in the arithmetic functioning of the system: in base 7, seven is written 10. However, on the symbolic and metrological plane, 84 represents the fulfilled seven — that is, the heptimal cycle fully deployed across the twelve internal subdivisions of each unit.
The Number 84
The number 84 has a special function in the symbolic system of Solaria. In decimal notation, 84 corresponds to 7 × 12. It therefore unites the heptimal cycle of seven with the duodecimal articulation of the unit.
In base 7, 84 is written 150. This form is significant because 15 in base 7 corresponds to 12 in decimal, while 10 in base 7 corresponds to 7 in decimal. The notation 150 can therefore be read as twelve brought into the cycle of seven.
The number 84 does not simply represent a quantity, but the complete cycle considered in its internal articulation. If seven is the cycle, 84 is the deployed cycle: seven units, each composed of twelve parts.
For this reason, alongside its standard positional form 150, the number 84 also possesses an autonomous symbolic form: the circle. The circle is not an additional digit and is not used in ordinary operations. It is an emblematic sign reserved for the number 84 when it appears as a symbol of completeness and totality.
The distinction between the two forms is important: 150 is the positional notation of the number, while the circle is its symbolic form.
Operations
Since the system is positional, it allows all ordinary arithmetic operations: addition, subtraction, multiplication, division, powers, roots, and fractions. Operations on pure numbers require no special rules beyond those of any other numerical base; only the value of the base and the point at which carries and borrows occur will change.
In addition and multiplication, the carry occurs upon reaching seven, not ten. In subtraction, when borrowing from a higher position, the borrowed unit is worth seven units of the lower position. Divisions and fractions function normally, although some fractions that are finite in decimal may become repeating in base 7, and vice versa.
In operations on measures articulated in units and internal parts, a dual carry logic applies instead. Internal parts transition to the unit when they reach twelve parts, that is, 15 in heptimal notation. Complete units transition to the next order when they reach seven units, that is, 10 in heptimal notation.
This dual logic does not make the system incoherent. It is the same principle that allows durations expressed in hours, minutes, and seconds to be added: the numbers remain calculable, but each level of the measure has its own transition point.
The symbolic notation of the circle for 84 does not modify calculation. In ordinary computations, 84 is written as 150 in base 7. The circle belongs to the symbolic plane, not the operative one.
Symbolic Value
The Solarian number system connects number, form, and cycle. The 2 and 5 represent the fundamental mirror-image pair: two elements in dialogue, which can be read as human and nature, or as the relationship between beings, forces, communities, or complementary principles.
The 0 represents their union. Although it has an ordinary mathematical function within the positional system, on the symbolic plane it does not indicate a simple absence, but the point at which duality is recomposed. For this reason it is associated with truth, harmony, and completion.
Seven represents the complete cycle. In positional notation it does not appear as an autonomous digit, but as the transition to 10. On the symbolic plane, however, seven is not merely a quantity: it is the principle of the cycle that completes itself and passes to a higher order.
This cycle is fully deployed in the number 84, formed by 7 units of 12 internal parts. For this reason, 84 represents the fulfilled seven — that is, the heptimal cycle made complete by its duodecimal articulation. Its ordinary notation is 150 in base 7, while its symbolic form is the circle.