The key to building an accurate honeycomb pattern is to measure the layout from the centers of the shapes, not from their edges. Once the center-to-center spacing is defined, every staggered row can be positioned using two simple formulas.
This method works for hexagons, circles, holes, and other repeated shapes arranged on a triangular grid. It can be used in most CAD and 3D modeling applications, including Plasticity.
Honeycomb Spacing and Offset Formulas
A honeycomb pattern is built from staggered rows. The spacing should be measured from the center of one shape to the center of the next, not from their edges.
Let S be the nearest-neighbor center-to-center distance. In the orientation shown here, it is also the horizontal spacing between adjacent centers in the same row:
S = horizontal center-to-center spacing
To position the next staggered row, use the following offsets:
X = S / 2
Y = (√3 / 2) × S ≈ 0.866 × S
X is the horizontal offset of the staggered row, while Y is the vertical center-to-center distance between rows.
Center spacing is not the same as the visible edge-to-edge gap. The size of the shapes controls the visible gap, while S controls the underlying grid.
For circular holes with diameter D, the horizontal edge-to-edge gap is:
gap = S − D
The shapes may touch, overlap, or remain separated, but their center positions are still calculated using the same grid formulas.
Once the second row is positioned, the two rows can be treated as a repeating module and duplicated to build the complete pattern.
Why Is the Vertical Factor 0.866?
The value 0.866 comes from the geometry of an equilateral triangle.
When the centers of three neighboring shapes are connected, they form an equilateral triangle whose side length is equal to the nearest-neighbor center spacing S.
The height of an equilateral triangle is:
h = (√3 / 2) × S
Because the vertical distance between two staggered rows is equal to this height:
Y = (√3 / 2) × S
Since √3 / 2 ≈ 0.8660254, the formula can be written as:
Y ≈ 0.866 × S
The value 0.866 is sufficiently accurate for most CAD and 3D modeling tasks. For very large arrays or precision-sensitive manufacturing work, use additional decimal places or the exact expression √3 / 2.
Honeycomb Pattern Example in Plasticity
Calculate and Position the Second Row
Suppose the horizontal center-to-center spacing is:
S = 10 mm
The horizontal offset of the second row is:
X = 10 / 2 = 5 mm
The vertical spacing between the rows is:
Y = (√3 / 2) × 10 ≈ 8.66 mm
In Plasticity, create the first row, duplicate it, and move the second row by 5 mm horizontally and 8.66 mm vertically.
The row may be moved upward or downward depending on the orientation of the model. The important part is to keep the same center-to-center offsets.
Build the Pattern with Two Array Directions in Plasticity
You can also build the honeycomb pattern directly from a single hexagon without calculating the horizontal and vertical offsets separately.
Start by selecting one hexagon and creating an array along a diagonal direction. Set the spacing to the required center-to-center distance between neighboring hexagons:
S = 10 mm
The diagonal array creates the first sequence of equally spaced hexagons along one axis of the triangular grid.
Next, select the resulting diagonal row and create a second array in the side direction. Use the same center-to-center spacing:
S = 10 mm
Because both array directions follow the axes of the triangular grid, the staggered honeycomb layout is produced automatically. There is no need to enter X = S / 2 and Y ≈ 0.866 × S as separate offsets.
Make sure the two array directions follow the center-to-center lines between neighboring hexagons. The angle between the directions should match the geometry of the triangular grid.
This method is useful when the CAD tool allows the array direction to be defined visually from the model geometry. It produces the same center layout as the formula-based method, but uses directional spacing instead of separate horizontal and vertical components.
Where This Spacing Method Works
The 0.866 spacing rule describes the arrangement of center points on a triangular grid, not the outlines of the objects.
This means the same method works for hexagons, circles, ventilation holes, cylinders, decorative cutouts, and other repeated features.
As long as the centers follow a triangular or hexagonal grid, use:
X = S / 2
Y = (√3 / 2) × S ≈ 0.866 × S
The objects may touch, overlap, or remain separated depending on their dimensions. Changing the size of the objects affects the visible gap, but it does not change the center-position formulas.
A square or rectangular grid works differently. Its rows and columns align directly, so no half-step horizontal offset is required:
X = 0
The vertical row spacing in a square grid is chosen independently according to the design requirements.
Use the 0.866 factor only for a staggered triangular arrangement, not for an ordinary square or rectangular array.
Frequently Asked Questions (FAQ)
What does S represent in a honeycomb pattern?
S is the nearest-neighbor center-to-center distance. In the orientation shown in this article, it is also the horizontal spacing between two adjacent centers in the same row.
What offsets should I use for the second row?
Use X = S / 2 for the horizontal offset and Y = (√3 / 2) × S ≈ 0.866 × S for the vertical spacing.
Why is the vertical spacing multiplied by 0.866?
The value 0.866 is an approximation of √3 / 2. This is the height-to-side ratio of the equilateral triangle formed by three neighboring center points.
Does this formula work only for hexagons?
No. It also works for circles, holes, cylinders, cutouts, and other repeated shapes, provided that their centers follow a triangular grid.
Is 0.866 accurate enough for large arrays?
It is accurate enough for most CAD and 3D modeling tasks. For large arrays or precision-sensitive manufacturing work, use 0.8660254 or the exact expression √3 / 2.
Can I build the pattern with an array tool?
Yes. You can either create and repeat a two-row staggered module or build the pattern from a single shape using two sequential arrays along the triangular grid directions.
How is a triangular grid different from a square grid?
In a triangular grid, every alternate row is shifted horizontally by S / 2, and the vertical spacing is approximately 0.866 × S. In a square grid, the rows align directly and the horizontal offset is 0.
Key Takeaway
Define the nearest-neighbor center spacing S, then position every alternate row using:
X = S / 2
Y = (√3 / 2) × S ≈ 0.866 × S
For example, when S = 10 mm, the second row is offset by 5 mm horizontally and approximately 8.66 mm vertically.
You can build the pattern either by positioning a staggered second row with the calculated offsets or by creating two sequential arrays along the triangular grid directions. The object size controls the visible gap, while the center spacing controls the underlying grid.
