Figure 1.
Developing embryonic leaves of Acer pseudoplatanus.
For such opposite decussate phillotaxy, a pair of symmetric leaves develop simultaneously. The primordia first expand over the stem apex (A) and then the two symmetric leaves meet (B), limiting each other in their future growth (C–E): a first lateral fold has appeared between the central and the lateral veins (C), a second one (D), and a secondary vein on the central one (E). Pictures A, B and D are MEB pictures courtesy of Isabel Le Disquet from IFR 83 of UPMC-Paris 6. Pictures C and E are optical microscope pictures.
Figure 2.
A mature leaf of Acer pseudoplatanus.
The largest lobe corresponds to the central vein (CV), while the lateral lobes develop around other major lateral veins (LV), radiating from the end of the petiole (P). Secondary lobes correspond to the end tip of some secondary veins (SV). There can be rare and small third order lobes around some end tip of third order veins (TV).
Figure 3.
Folded immature leaf extracted from a bud of Acer campestre.
A: The abaxial side shows the anticlinal folds (arrows) running along veins and ending at peaks (circles). B: The adaxial side shows the synclinal folds (arrows) running along “anti-veins” and ending at sinuses (circles). Peaks and sinuses stand in the contact plane of the pair of leaves (figure 1E), but while peaks are at the extreme of this contact surface, sinuses are inside.
Figure 4.
Side and front views of Kirigami leaves.
The side view shows the correspondence of anticlinal folds with the main veins, the front view shows the adaxial contact plane, where the leaf perimeter is located. A: Ribes nigrum (Saxifragales), B: Pelargonium cuculatum (Geraniales), C: Malva sylvestris (Malvales).
Figure 5.
Local relationship between synclinal folds and sinuses on immature leaves before their expansion outside the bud.
Synclinal fold angles φ are measured from side views on folded leaves extracted from buds (A and C). Leaves are then unfolded to measure the corresponding sinus opening angle ψ from a top view (turning the back in b, and turning the top in d). The fold, setting the sinus whose contours locally superimpose, is the symmetry axis of the sinus. If the leaf is a folded flat surface, this symmetry simply makes ψ = 360° − 2 φ, as sketch in B and D. This relation (line) is checked for many folds on different leaves and species. The fact that it works shows that even if the leaves grow folded in the bud (figure 3), they are already flat. The plot also shows the large range of angle variation within one species and for all the species. Pictures are two extreme cases: when the fold angle become close to 90° the valley disappears (Malva, left), while when it becomes close to 180°, the valley becomes a simple cut (Tetrapanax, right). The studied palmate leaf species belong to different order of eudicots. Following the APG II classification Acer pseudoplatanus and Acer campestre (Sapindales), Malva sylvestris and Alchemilla vulgaris (Malvales) are Rosids, Tetrapanax papyriferum (Apiales) is an Asterid and Gunnera manicata is a Gunnerale.
Figure 6.
Mature leaves of different species, numerically folded back.
The leaf contour is pink. Primary and secondary veins are respectively blue and green. Primary and secondary anti-veins are yellow and red respectively. Only veins ending at peaks are represented and stand for anticlinal folds along segments linking two consecutive branching points or a branching point to a peak. Synclinal folds run along segments (anti-veins) linking a sinus to the branching point of the two surrounding veins. The thickness of the leaf is not taken into account and the leaf is folded back onto a plane, holding the angles to the best (see material and methods, figure 13). A: Acer pseudoplatanus, B: Malva sylvestris, C: Ribes nigrum, D: Sida hermaphrodita, E: Gunnera manicata. Following the APG II classification, these species belong to different orders of core eudicots: Acer pseudoplatanus (Sapindales), Sida hermaphrodita and Malva sylvestris (Malvales) are Rosids, Gunnera manicata is a Gunnerale and Ribes nigrum is a Saxifragale.
Figure 7.
Acer pseudoplatanus leaves: pictures and numerical folding (same representation as figure 6).
A: One lobe leaf, with only one anticlinal fold. B: Three lobe leaf, with two lateral anticlinal and three synclinal folds. C: Five lobe leaf, with four lateral anticlinal and five synclinal folds, and several secondary folds (as in figure 6A). The secondary folding is necessary here to obtain a refolding of the contour on a simple curve (see figure 13). One lobe leaves are found on new sprouts.
Figure 8.
Geometric relationships between two successive lobes and sinuses coming from the Kirigami property.
A: Two consecutive primary lobes have veins of lengths Ra and Rc. They are respectively making an angle α and β with the anti-vein, of length Rb, between them. B: The vein of length Rc is surrounded by two anti-veins of lengths Rb and Rd. These are respectively making an angle β and γ with the vein. C: According to A where the folded “leaf” contour is the same for both sides of the fold, it is possible to compute Rc from Ra, Rb, α and β (see formula and figure 14 in methods). For β growing from 0 to α, Rc simply run along the perimeter, growing from Rb (left inset) to Ra (right insert). To compare leaf of different sizes, we plot Rc divided by Ra, compared with the Kirigami prediction. D: Similarly, for tow anti-veins surrounding a vein, when β grow from 0 to γ, Rb decrease from Rc (right inset) to to Rd (right inset). As the anti-veins were not ordered by size (contrary to the veins), the figure is symmetric. Points represent 121 sycamore leaves. For each leaf, all pair of consecutive primary veins or anti-veins (four for the leaf presented) have been measured.
Figure 9.
Example of an enrolled leaf: the Philodendron bipenifolium (Araceae).
A: A juvenil Philodendron bipenifolium leaf. B: Detail of the enrolled leaf. The leaf does not grow folded but enrolled; it is an involute leaf. C: A mature Philodendron bipenifolium leaf. D: The same leaf numerically folded. Like for folding of figure 6, the thickness of the leaf is not taken into account (see figure 13 and material and methods for a detailed explanation). As the leaf does not grow folded, it does not obey the Kirigami property and is not foldable with its contour lying on a simple curve.
Figure 10.
Geometric relationships between two successive lobes and sinuses coming from the Kirigami property, for the enrolled Philodendron bipenifolium leaf.
Same notations and plots as in figure 8. A: Two consecutive primary lobes have veins of lengths Ra and Rc. They are respectively making an angle α and β with the anti-vein, of length Rb, between them. B: The vein of length Rc is surrounded by two anti-veins of lengths Rb and Rd. These are respectively making angle an β and γ with the vein. As in figure 8, the predictions of the Kirigami property are compared to the measured values for veins (C) and for anti-veins (D). Points represent 85 Philodendron bipenifolium leaves. For each leaf, all pair of consecutive primary veins or anti-veins (four for the presented leaf) have been measured. One observes that points are scattered in both figures. This figure, together with the previous one, shows that the Kirigami property does not appear for any shape.
Figure 11.
Simplified relationship on two successive lobes and sinuses that can be used to recognize the Kirigami property.
As in figure 8, the graphs express the relation between the folds length and the angles. A: Length ratio (Ra/Rc) of two consecutive main veins in function of the difference (α–β) between the angles there are making with the anti-vein. The smaller the angle, the smaller the vein. If the anti-vein (sinus) is at the middle between the two veins, then the two veins have the same length (ratio 1, right inset). On the contrary, if the angle for the second lobe becomes small, then the length of the second lobe should be smaller, eventually becoming equal to the length of the anti-vein or sinus (left inset). B: Length ratio (Rb/Rd) of two consecutive main anti-veins in function of the difference (β–γ) between the angles with the vein. Similarly, the relation express that the smaller the angle, the longer the anti-vein. When the angles are equal the two anti-veins have the same length (right inset), while if one angle becomes smaller, the corresponding anti-vein becomes longer, eventually becoming equal to the vein (left inset). Contrary to figure 8, these relations are approximate, scattering the points. They are exact only if the opening angle of the lobes is constant, which is a first approximation, but in practice mixes different relationships for each lobe opening angle. However, they are simple way to judge of the Kirigami property, judging by the eye the ratio of length and angle difference, that the anti-veins and veins are axes of symmetry of the contour. For instance they are clearly wrong for figure 10B (α is smaller than β, but Ra is much longer than Rc).
Figure 12.
Dissected Kirigami buds showing the packed folded leaves.
One leaf can be limited by an other leaf of the same age (opposite decussate phyllotaxy, A, while there are also two other pairs of leaves, older and younger, delimiting the volume); or leaves of different ages (spiral phyllotaxy), either protected by an envelope, B, or not, C. A: Acer pseudoplatanus (Sapindales), B: Murus platanifolium (Rosales), C: Pelargonium cuculatum (Geraniales). In all cases, the leaf fit the available volume, and no space is left free in the bud.
Figure 13.
A: On a leaf, the first order vein (blue line) of a maple leaf and secondary vein (green line) are drawn. The first order anti-vein (orange line) and the second order anti-veins (red line), always going from the intersection of two vein up to the sinus between the lobes, are also drawn. The contour of the leaf (mangenta line) is numerically detected. B: The result with the contour and only the main veins and anti-veins, and the first angles between them. C: Half of the precedent sketch once refolded. α and γ are reversed, and their respective contour drawn inversed. D: The whole refolded leaf, using only its main folds. E: Sketch of the leaf with all its folds: main and secondary ones. F: Scheme of a secondary fold, unfolded, and, G: folded in a plane. The news angles are obtained as described in the text. H: The whole set of veins and anti-veins is drawn, with their respective contour, giving the completely refolded leaf.
Figure 14.
A: If the tips of Ra, Rb, Rc align then the two vectors (Rc –Rb) and (Ra –Rc) are collinear. Their cross product is null. B: For a given set of α, β, Ra, Rb, with this property, one can deduce Rc.