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Fig. 3 in Repeatability Analysis Of Egg Shape In A Wild Tree Sparrow (Passer Montanus) Population: A Sensitive Method For Egg Shape Description

Fig. 3. The effect of egg-photographing on the description of outline. Panel a shows ten outlines described following the photos of ten randomly chosen eggs, panel b shows ten outlines described fol-

Figure 16 in Mathematical interpretation of avian egg shapes

Figure 16. Ovoid curves: a) Hügelschäffer (according to Obradovic et al., 2013); b) Blaschke (according to Köller, 2000); с) Schauberger (according to Coats, 2001).

Figure 15 in Mathematical interpretation of avian egg shapes

Figure 15. Yamamoto oval (а) (according to Yamamoto, 2020); Cartesian oval (b) (according to Köller, 2000); Möller oval (c) (according to Möller, 2009).

Figure 13 in Mathematical interpretation of avian egg shapes

Figure 13. Cone-like ovoids: a) zeta-curve (Stadnicki, 2015); b) Phalacrocorax carbo; c) Anser anser; d) Phalacrocorax pelagicus; e) Grus canadensis; f) Grus grus.

Figure 14 in Mathematical interpretation of avian egg shapes

Figure 14. Cartesian ovals: а) with p = 1, q = 2 and c = 5; b) with p = 1, q = 3 and c = 5 (Beverlin, 2006); с) hyperbolic cone (Kirsh, 2010).

Fig. 10 in Mathematical interpretation of avian egg shapes

Fig. 10. Spher-obtuse аsymmetric pseudo-ovoid: 1, 9, 17) construction of ovoids in the shape matrix; 2, 10, 18) egg profile diagram; bird egg profiles: 3) Accipiter gentilis; 4) Accipiter nisus; 5) Falco vespertinus; 6) Falco tinnunculus; 7) Panurus biarmicus; 8) Asio otus; 11) Milvus milvus; 12) Hieraaetus pennatus; 13) Picus canus; 14) Falco cherrug; 15) Columba oenas; 16) Falco tinnunculus; 19) Dendrocopos major; 20) Cuculus canorus; 21) Sylvia atricapilla; 22) Fringilla coelebs; 23) Crex crex; 24) Phasianus colchicus.

Fig. 9 in Mathematical interpretation of avian egg shapes

Fig. 9. Аsymmetric sphere-rounded pseudo-ovoid: 1, 9, 17) construction of ovoids in the shape matrix; 2, 10, 18) egg profile diagram; bird egg profiles: 3) Egretta alba; 4) Podiceps nigricollis; 5) Porzana parva; 6) Ardea cinerea; 7) Cygnus olor; 8) Podiceps cristatus; 11) Cygnus olor; 12) Casuarius casuarius; 13) Egretta garzetta; 14) Mergus serrator; 15) Egretta alba; 16) Phalacrocorax carbo; 19) Anser anser; 20) Pelecanus crispus; 21) Podiceps grisegena; 22) Cygnus olor; 23) Phalacrocorax carbo; 24) Podiceps cristatus,

Figure 12 in Mathematical interpretation of avian egg shapes

Figure 12. Rounded -drop-shaped asymmetric pseudo-ovoid: 1, 9, 17) construction of ovoids in the matrix of forms; 2, 10, 18) egg profile diagram; bird egg profiles: 3) Oriolus oriolus; 4) Podiceps nigricollis; 5) Remiz pendulinus; 6) Porzana porzana; 7) Procellaria leucomelas; 8) Fulica atra;11) Catharacta maccormicki; 12) Corvus corax; 13) Phalacrocorax pelagicus; 14) Riparia riparia; 15) Larus genei; 16) Alauda arvensis; 19) Platalea leucorodia; 20) Upupa epops; 21) Larus cachinnans; 22) Pica pica; 23) Lunda cirrhata; 24) Gavia arctica.

Fig. 11 in Mathematical interpretation of avian egg shapes

Fig. 11. Asymmetric rounded-typical pseudo-ovoids: 1, 9, 17) construction of ovoids in the shape matrix; 2), 10), 18) egg profile diagram; bird egg profiles: 3) Fringilla coelebs; 4) Tetrastes bonasia; 5) Chloris chloris; 6) Fulica atra; 7) Sturnus vulgaris; 8) Turdus merula; 11) Acrocephalus palustris; 12) Turdus viscivorus; 13) Cuculus canorus; 14) Oriolus oriolus; 15) Luscinia luscinia; 16) Dendrocopos major; 19) Haematopus ostralegus; 20) Remiz pendulinus; 21) Porzana porzana; 22) Anthropoides virgo; 23) Cygnus cygnus;24) Anas strepera.

Figure 4 in Mathematical interpretation of avian egg shapes

Figure 4. Rounded symmetric pseudo-ovoid: 1, 9, 17) construction of ovoids in the shape matrix; 2, 10, 18) egg profile diagram; bird egg profiles: 3) Pygoscelis papua; 4) Struthio camelus; 5) Alcedo atthis; 6) Strix aluco; 7) Hieraaetus pennatus; 8) Aegolius funereus; 11) Pygoscelis antarctica; 12) Merops apiaster; 13) Strix aluco; 14) Ardeola ralloides; 15) Streptopelia decaocto; 16) Asio otus; 19) Bubo bubo; 20) Ciconia ciconia; 21) Columba palumbus; 22) Streptopelia turtur; 23) Anas platyrhynchos; 24) Ixobrychus minutus.

Figure 5 in Mathematical interpretation of avian egg shapes

Figure 5. Obtuse symmetric pseudo-ovoid: 1, 9, 17) construction of ovoids in the shape matrix; 2, 10, 18) egg profile diagram and profiles: 3) Columba livia; 4) Anas platyrhynchos; 5) Egretta alba; 6) Ixobrychus minutus; 7) Columba palumbus; 8) Rhea americana; 11) Otis tarda; 12) Egretta alba; 13) Merops apiaster; 14) Aythya ferina; 15) Caprimulgus europaeus; 16) Anas quеrquedula; 19) Upupa epops; 20) Egretta alba; 21) Anser anser; 22) Egretta garzetta; 23) Ardea cinerea; 24) Ardea cinerea.

Figure 3 in Mathematical interpretation of avian egg shapes

Figure 3. Sphere-like symmetric pseudo-ovoid: 1) plotting of ovoids in the shape matrix; 2) a profile for comparison with real egg shapes; bird egg profiles: 3) Otus scops; 4) O. brucei; 5) Athene noctua; 6) Merops apiaster; 7) M. superciliosus; 8) Alcedo atthis.

Figure 2 in Mathematical interpretation of avian egg shapes

Figure 2. Variability of the polar zones curvature of symmetric pseudo-ovoids: а) short, b) normal, c) long.

Figure 1 in Mathematical interpretation of avian egg shapes

Figure 1. The formation of asymmetric and symmetric (а) pseudo-ovoids; b-с) schemes for measuring the semi-axes: (L – length, D – diameter, rc – cloacal and ri – infundibular radii, ip – interpolar segment, lc - cloacal and li - infundibular semi-axes)

Figure 8 in Discrete aggregate analysis of ovoid egg shapes in various bird species

Figure 8. Golden proportion manifested in the shape of bird eggs: 1, 13, 25, 37, ovoid profiles; 2, 14, 26, 38, ovoid profile profiles for comparison with bird eggs. Bird eggs: 3 – Alectoris chukar; 4 – Glareola pratincola; 5 – Sturnus vulgaris; 6 – Sylvia nisoria; 7 – Milvus milvus; 8 – Buteo buteo; 9 – Falco cherrug; 10 – Lanius collurio; 11 - Perdix perdix; 12 – Garrulus glandarius; 15 – Sylvia communis; 16 – Perdix perdix; 17 – Numida meleagris; 18 – Alectoris chukar; 19 – Garrulus glandarius; 20 – Coccothraustes coccothraustes; 21 – Larus melanocephalus; 22 – Dendrocopos major; 23 – Picus canus; 24 – Jynx torquilla; 27 – Luscinia svecica; 28 – Oenanthe oenanthe; 29 – Turdus merula; 30– Picus canus; 31– Aquila pomarina; 32 – Perdix perdix; 33 – Lanius collurio; 34 – Anthus trivialis; 35 – Jynx torquilla; 36– Hieraaetus pennatus; 39 – Turdus philomelos; 40 – Oriolus oriolus; 41 – Ficedula albicollis; 42 – Dendrocopos leucotos; 43 – Delichon urbica; 44 – Coccothraustes coccothraustes; 45 – Garrulus glandarius; 46 – Corvus frugilegus; 47 – Turdus merula; 48 – Gallinula chloropus.

Figure 5 in Discrete aggregate analysis of ovoid egg shapes in various bird species

Figure 5. Comparison of geometric standards (1, 7, 13, 19, 25, 31) with profiles of real eggs: 2 – Athene noctua; 3 – Neophron percnopterus; 4 – Merops apiaster; 5 – Phylloscopus sibilatrix; 6 – Alcedo atthis; 8 – Falco vespertinus; 9 – Aegithalos caudatus; 10 – Merops apiaster; 11 – Ficedula hypoleuca; 12 – Jynx torquilla; 14 – Sterna hirundo; 15 – Buteo buteo; 16 – Falco cherrug; 17 – Upupa epops; 18 – Dendrocopos major; 20 – Picus canus; 21 – Perdix perdix; 22 – Phasianus colchicus; 23 – Corvus monedula; 24 – Coccothraustes coccothraustes; 26 – Limosa limosa; 27 – Pica pica; 28 – Luscinia luscinia; 29 – Corvus frugilegus; 30 – Alca torda; 32 – Uria aalge; 33 – Numenius arquata; 34 – Recurvirostra avosetta; 35 – Alca torda; 36 – Corvus corax.

Figure 3 in Discrete aggregate analysis of ovoid egg shapes in various bird species

Figure 3. Discrete variants of combinations of the lateral arcs of the ovoid profiles (matrices): a - 0.75D; b -1.0 D; c - 1.25D; d - 1.5D; e - 1.75D.

Figure 7 in Discrete aggregate analysis of ovoid egg shapes in various bird species

Figure 7. Construction of ovoid profiles with golden ratio: a - construction of ovoids with doubling of radii of circles; b - construction of ovoids with increasing of radii of circles by F=1,618; c - getting golden section by a combination of four circles; d - construction of ovoids with golden section by pentagons.

Figure 11 in Discrete aggregate analysis of ovoid egg shapes in various bird species

Figure 11. Schematics for calculating the complementarity indices of avian eggs: 1, 2, 9, 10 - scheme and geometric profile of eggs with a complementarity index close to unity (rc=ri=(L–D)/2); 3 – Podiceps cristatus; 4 – Cygnus olor; 5 – Egretta alba; 6 – Pelecanus crispus; 7 – Egretta garzetta; 8 – Mergus serrator; 11 – Anser anser; 12 – Casuarius casuarius; 13 – Porzana parva; 14 – Phalacrocorax carbo; 15 – Podiceps nigricollis; 16 – Ardea cinerea; 17, 18 - scheme and geometric profile of eggs with a complementarity index close to 1,333 (rc=ri=lipz=L/3); 19 – Pygoscelis papua; 20 – Nyctea scandiaca; 21 – Strutio camelus; 22 – Dendrocopos minor; 23 – Alcedo atthis; 24 – Buteo chemilasius; 25 – Jynx torquilla; 26 – Asio otus; 27 – Streptopelia decaocto; 28 – Merops apiaster; 29 – Asio flammeus; 30 – Aquila pomarina; 31 – Bubo bubo; 32 – Columba oenas; 33, 34 – scheme and geometric profile of eggs with a complementarity index greater than 1,5 (rc=ri/lipz˃1,5); 35 – Athene noctua; 36 – Otus scops; 37 – Otus brucei; 38 – Merops apiaster; 39 – Strix aluco; 40 – Pygoscelis papua; 41 – Alcedo atthis; 42 – Merops superciliosus; 43 – Milvus migrans; 44 – Asio flammeus; 45 – Strutio camelus; 46 – Hieraaetus pennatus; 47 – Strix aluco; 48 – Accipiter nisus.

Figure 2 in Discrete aggregate analysis of ovoid egg shapes in various bird species

Figure 2. Relationship between the ovoid constructs with the diagonals of the square and the double square.