1

Crystal Chemistry of Borates: the Classification and Algebraic

Description by Topological Types of Fundamental Building Blocks

Guanghui Yuan and Dongfeng Xue* State Key Laboratory of Fine Chemicals, Department of Materials Science and Chemical Engineering, School of Chemical Engineering, Dalian University of

Technology, 158 Zhongshan Road, Dalian 116012, China

* Corresponding author. E-mail: dfxue@chem.dlut.edu.cn

Supplementary

545 anhydrous borates and 296 hydrated borates are statisticed and classified into 6

kinds (single ∆ or T borates; branched borates; normal-ring borates; bridge-ring

borates; ‘8’-shaped-ring borates and combined-ring borates) based on the different

topological structures of their FBBs. Subsequent division related on the linkages

(isolate, chains, layers and networks) between these FBBs in borate structures. On the

basis of the two restrictive factors, all borates (including boron minerals and synthetic

borates) can be classified into 22 kinds, as shown in Figure 5 in article.

It should be noted that the FBBs of single ∆ or T borates are impossible to

polymerize layers or network. Because besides including the information of the

fundamental building block, the FBB should reflect the structure hypostasis of the

polyanions. When polymerizing layers or network, the anions must be connected

together in many ways and induce the FBB to become more complex structure block

not single ∆ or T. For example, although single BO4 (T) can repeat the polyanions in

LaAl2.03(B4O10)O0.54 (ICSD 59318), it couldn’t indicate the ring structure (Figure S1).

Therefore, the FBB of this compound is <6T> not T, and this compound is classified

as normal-ring layer borate with descriptor: 6: ∞2[<6T>].

2

a

(a)

(b)

Figure S1

(a) Two-dimensional anion-layer in the structure of LaAl2.03 (B4O10) O0.54. All the La

atoms and Al atoms are omitted for clarity. (b) FBB structure and FBB topology in

LaAl2.03 (B4O10) O0.54.

3

Tables S1 to S6 are for anhydrous borates and Tables S1’ to S6’ are for hydrated

borates. Some hypothesizes and treatments should be pointed out:

(1) Some multi-borates such as borate-silicates or borate-arsenates are ignored,

because in these compounds the bond valences of a bond Si-O in SiO4 or

As-O in AsO4 are higher than that of B-O in BO3 or BO4 structural unit. The

structural characteristics are dominated by the SiO4 or AsO4 structural units

and less by the borate units.

(2) In borate-phosphates, PO4 (T’) is treated as BO4 (T) because many

borate-phosphates (ICSD 409327, 409459, 409515, etc.) are synthesized and

their properties especially the nonlinear optical property are similar to borates,

furthermore, P atom can only form PO4 with O atom, which are stable

relatively.

(3) As for other multi-anion borates only poly-borate anions are considered.

(4) Some new borates, which are found or synthetic after ICSD (2004), are also

listed in these tables.

Table S1 Anhydrous borates with the Single BO3 or BO4 FBB

Chemical Formula FBB R Crystal

System

Centro

symmetric

Note

LiGeBO4 T 0 Orthorhombic No

KGe2BO6 T 0 Orthorhombic No

α-Ag3BO3 ∆ 0 Trigonal No

K2LiBO3 ∆ 0 Monoclinic No

CaM(BO3)O ∆ 0 Orthorhombic No M=Al, Ga

BaRECu (BO3)O2 ∆ 0 Tetragonal No RE=La, Nd

MBe2BO3F2 ∆ 0 Monoclinic No M=Cs, Rb

KBe2BO3F2 ∆ 0 Trigonal No

M5BO9 ∆ 0 Orthorhombic No M5=Al5, Al4.5Cr0.5, Al4.8Cr0.2

(Ln)3WBO9 ∆ 0 Hexagonal No Ln=La, Eu

BaZr (BO3)2 ∆ 0 Trigonal No

BaZn2(BO3)2 ∆ 0 Orthorhombic No

4

MCu2 (BO3)2 ∆ 0 Tetragonal No M=Sr, Sr0.74Ba0.27, Sr0.66Ca0.34

MgBa2 (BO3)2 ∆ 0 Trigonal No

K3M3(BO3)2O6 ∆ 0 Hexagonal No M=Ta, Nb

Sr2Be2B2O7 ∆ 0 Hexagonal No

K2Al2B2O7 ∆ 0 Trigonal No

SrAl2B2O7 ∆ 0 Trigonal No

Ba3Ln(BO3)3 ∆ 0 Hexagonal No Ln=Lu, Ho, Tm, Yb, Sc

RECa4(BO3)3O ∆ 0 Monoclinic No RE=Gd, Sm, Tb, Lu, La, Nd,

Er, Y

Ca3(BO3)3F ∆ 0 Monoclinic No

MM’3(BO3)4 ∆ 0 Trigonal No MM’=YbAl, NdGa, NdAl,

CeSc, YAl, (Y0.5Eu0.5)Al,

CeSc, (Y0.9Eu0.1)Al,

(Y0.5Be0.5)Fe, LaFe, NdFe,

TbFe, (Nb0.5Ti0.5)Fe,

Y(Al0.5Ga0.5), β-LaSc

REAl3(BO3)4 ∆ 0 Monoclinic No RE=Gd, Yb, Gd0.97Eu0.03

Sr3(RE)2(BO3)4 ∆ 0 Orthorhombic No RE=Er, Nd, Pr, La

Pb3B4O9 ∆ 0 Trigonal No

Ca3La3(BO3)5 ∆ 0 Hexagonal No

Na3La9(BO3)8O3 ∆ 0 Hexagonal No

MBO3 ∆ 0 Trigonal Yes M=Al, Nd, Eu, Fe, In, Dy, Er,

Ti, Sm, Sc, (Sc0.75Tb0.25)

LnBO3 ∆ 0 Hexagonal Yes Ln=Gd, Ho, Er, Tm

M3BO3 ∆ 0 Monoclinic Yes M=Li, Na

LaBO3 ∆ 0 Monoclinic Yes

MZn4BO3 ∆ 0 Monoclinic Yes M=K, Rb

β-Ag3BO3 ∆ 0 Trigonal Yes

M3PBO3 ∆ 0 Hexagonal Yes M=Sr, Ba

CrBO3 ∆ 0 Orthorhombic Yes

REBO3 ∆ 0 Trigonal Yes RE=Yb, Lu, V

MM’BO3 ∆ 0 Monoclinic Yes MM’=LiCo, NaBa, LiFe,

LiMg, LiBa, LiSr, LiMn,

LiZn, Li1.13Zn0.93

LiCdBO3 ∆ 0 Cubic Yes

MM’2BO3 ∆ 0 Orthorhombic Yes MM’2=KNa2, CsNa2, RbNa2,

KLi2

M2(BO3)O ∆ 0 Orthorhombic Yes M2= Mn2, Co1.5Zr0.5, α-PbAl,

β-PbAl, Co1.5Ti0.5,

Mg1.33Al0.21Fe0.12Ti0.34,

Mg1.1Fe0.64Ti0.16Al0.1

MM’(BO3)O ∆ 0 Orthorhombic Yes MM’=CaY, SrAl, MgSc,

MgGa, NiSc, FeMn, FeCo,

FeZn, PbGa,

5

Mg(Mg0.12Fe0.64Al0.1Ti0.16)

MM’(BO3)O ∆ 0 Monoclinic Yes MM’= Mn(Al0.5Y0.5),

(Mg0.76Mn0.24)Mn, FeFe, Li2Al

Be2BO3F ∆ 0 Monoclinic Yes

Ca2BO3Cl ∆ 0 Monoclinic Yes

Gd3(BO3)F3 ∆ 0 Monoclinic Yes

M2M’(BO3)O2 ∆ 0 Orthorhombic Yes M2M’=Ni2Cr, Cu2Al, Mn2Mn,

Ni2Al, Ni2Fe, Ni2Ga, Ni2V,

(Mg1.81Mn0.08Fe0.01)Mn,

Mg1.9(Mn0.91Sb0.19),

(Fe1.25Mg0.75)Fe,

(Mg1.85Fe0.15)(Fe0.60Al0.40),

(Mg1.92Fe0.0825)(Fe0.566Al0.41Ti0.02)

Mg1.93Fe0.967Al0.093,

Mg1.941Fe0.967Al0.083,

Mg1.966Fe0.911Al0.118,

Mg2.11Al0.31Fe0.53Ti0.05,

Mg2.33Al0.17Fe0.37Ti0.02Sb0.12,

Mg1.707Fe1.213Al0.08,

(Mg1.84Fe0.16)(Fe0.54Ti0.21Mg0.15Al0.10)

LiU(BO3)O2 ∆ 0 Monoclinic Yes

M3(BO3)O2 ∆ 0 Orthorhombic Yes M3=Co3, Co2.1Al0.9,

Mg1.71Mn1.29,

Mg1.33Mn1.49Sb0.17,

Mg1.93Mn1.07,

Mg1.98Fe0.02Mn0.78Fe0.2Al0.02,

Mg1.25Mn1.57Sb0.18,

Mg1.42Mn1.43Fe0.22Mn0.88

NaU(BO3)O2 ∆ 0 Orthorhombic Yes

Li0.31V0.69(BO3)O2 ∆ 0 Orthorhombic Yes

M2M'(BO3)O2 ∆ 0 Monoclinic Yes M2M’=Cu2Fe, Cu2Ga, Mn2Sr

M3(BO3)2 ∆ 0 Orthorhombic Yes M=Mg, Co, Ni, Mn, Sr, Cd

M3(BO3)2 ∆ 0 Trigonal Yes M=Ca, Hg, Eu

MM’(BO3)2 ∆ 0 Trigonal Yes MM’=SrSn, BaZr, MgSn,

CaSn, K2Zr,

(Mn0.83Fe0.15Ca0.02)Sn,

(Mn0.83Fe0.15Ca0.02)Sn

M2M’(BO3)2 ∆ 0 Monoclinic Yes M2M’=Sr2Cu, Ba2Ca, Be2Sr,

Zn2Zn

CsLi5(BO3)2 ∆ 0 Monoclinic Yes

MM’2(BO3)2 ∆ 0 Orthorhombic Yes MM’2=BaBe2, BaZn2, CuBa2

Li3RE(BO3)2 ∆ 0 Monoclinic Yes RE=In, Gd, Al

Li4Ca(BO3)2 ∆ 0 Orthorhombic Yes

M2M’2(BO3)2O ∆ 0 Trigonal Yes M2M’2=Na2Al2, K2Ga2,

6

Zn2Zn2

Na2Gd2(BO3)2O ∆ 0 Monoclinic Yes

Ca(RE)2 (BO3)2O ∆ 0 Trigonal Yes RE=Al, Eu

Mg3Ti (BO3)2O2 ∆ 0 Orthorhombic Yes

Sr2Al2 (BO3)2O2 ∆ 0 Orthorhombic Yes

Bi4(BO3)2O3 ∆ 0 Monoclinic Yes

M5M’(BO3)2O4 ∆ 0 Orthorhombic Yes M5M’=Co5Sn, Co5Mn, Cu5Sn,

Ni5Ti, Ni5Mn, Ni5Ge, Ni5Zr,

Ni5Sn, Ni5(Ni0.33Sb0.67),

Ni5(Ni0.33Ta0.67), Ni5Hf, Ni5V,

Zn5Mn

M3M’3(BO3)2O4 ∆ 0 Monoclinic Yes M3M’3=Nb3(Nb2.33Nb0.67),

Mg3Sn3

Nd2Sr3Cu3(BO3)2O6 ∆ 0 Tetragonal Yes

Ba3Ti3(BO3)2O6 ∆ 0 Hexagonal Yes

Ba2RE(BO3)2Cl ∆ 0 Monoclinic Yes RE=Yb, Ho

Li2Yb5(BO3)3O4 ∆ 0 Monoclinic Yes

LiLn6(BO3)3O5 ∆ 0 Monoclinic Yes Ln=Pr→Tm, Y

Li6RE(BO3)3 ∆ 0 Monoclinic Yes RE=Yb, Ho

Ba3Dy(BO3)3 ∆ 0 Trigonal Yes

MM’4 (BO3)3 ∆ 0 Cubic Yes MM’4=LiSr4, NaSr4, NaBa4

Sr3Sc(BO3)3 ∆ 0 Trigonal Yes

Sr5(BO3)3F ∆ 0 Orthorhombic Yes

Cu2Al6(BO3)4O5 ∆ 0 Tetragonal Yes

Ba3(RE)2(BO3)4 ∆ 0 Orthorhombic Yes RE=La, Pr, Nd

MM’3 (BO3)4 ∆ 0 Monoclinic Yes MM’3=LaSc3, NdAl3, PrSc3

Ba2Sc2B4O11 ∆ 0 Monoclinic Yes

Al6(BO3)5F3 ∆ 0 Hexagonal Yes

Sr6RESc(BO3)6 ∆ 0 Trigonal Yes RE=Ho, Y

La26B8O21 ∆ 0 Monoclinic Yes

Sb3Mn8.72Al0.28Mn21.4

Mg13.6(BO3)16O32 ∆ 0 Orthorhombic Yes

MBO4 T 0 Tetragonal Yes M=Nb, Nb0.53Ta0.47, Ta

AlMgBO4 T 0 Orthorhombic Yes

BaBOF3 T 0 Orthorhombic Yes F is treated as O

Al4Co(BO4)2O2 T 0 Monoclinic Yes

Ni2NbBO6 T 0 Orthorhombic Yes

(RE)3BO6 T 0 Orthorhombic Yes RE=Cr, Fe

Ni7U(BO4)4 T 0 Orthorhombic Yes

Na2AlAs4BO14 T 0 Monoclinic Yes

PbAsBO5 T 1 Trigonal No

BaAsBO5 T 1 Hexagonal Yes

MB2O4 ∆ 1 Orthorhombic Yes M=β-Ca, Sr, Eu

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Table S2 Anhydrous borates with the Branched FBB

Chemical

Formula

FBB R Crystal System Centro

symmetric

Note

BaCuB2O5 2∆ 0 Monoclinic No

MM’B2O6 2∆ 0 Monoclinic No MM’=RbNb, CsTa,

CsNb, TlNb

Cu(Cu1.63Zn0.37)

Se2B2O7 2T 0 Orthorhombic No

CaU2B2O10 2∆ 0 Monoclinic No

*Li3Eu2B3O9 2∆ & ∆ 0 Monoclinic No

Cr2BP3O12 ∆3 |T’| 0 Trigonal No

Pb6BP5O20 T 4 |T’| 0 Tetragonal No

*Y16.86B8O38 2∆ & ∆ 0 Monoclinic No

χ-LnBO3 ∆2T 0 Triclinic Yes Ln=Dy, Er

LnB2O4Cl 2∆ 0 Triclinic Yes Ln=La, Pr

MM’B2O5 2∆ 0 Triclinic Yes MM’=MgMn, MnCo,

Ni1.5Zn0.5, Co2, Cd2,

(β-Mg)2, (Co1.5Zn0.5)2

MB2O5 2∆ 0 Monoclinic Yes M=Th, U, Na4, Mg2,

Sr2, Ca2

MM’B2O5 2∆ 0 Monoclinic Yes MM’=CaMn, MgCa,

NaSc, LiAl

Er2B2O5Cl2 2∆ 0 Orthorhombic Yes

MCu2B2O6 2∆ 0 Monoclinic Yes M=Co, Cd

*Cu3B2O6 2∆ & ∆ 0 Triclinic Yes

Co5BP3O14 ∆T’ 0 Monoclinic Yes

MgUB2O7 2∆ 0 Orthorhombic Yes

MGa2B2O7 2∆ 0 Orthorhombic Yes M=Sr, Ba

M2M’M’’B4O10 2∆ 0 Monoclinic Yes M2M’M’’=Sr2LiSc,

Sr2LiIn,

Ca2(Mg0.86Fe0.14)(Mg0.92Fe0.08)

Ba5B4O10F2 2∆ 0 Monoclinic Yes

*Sr2Sc2B4O11 2∆ & ∆ 0 Triclinic Yes

*Ho8.66B4O19 2∆ & ∆ 0 Monoclinic Yes

Cu9Ti2B6O16 2∆ 0 Triclinic Yes

TlBO2 ∆T 1 Tetragonal No

CuLnB5O10 T 4 |∆| 1 Orthorhombic No Ln=Tb, Lu, Tm

CaBeB2O5 ∆2T∆ 1 Monoclinic Yes

Ba3BP3O12 T 3 |T’| 1 Orthorhombic Yes

BiB3O6 ∆T∆ 2 Monoclinic No

Li14BeB10O27 T 4 |∆| 2 Hexagonal Yes

Li4B7O12Cl ∆T∆ 3 Cubic No

NiGdB5O10 O 4 |T| 3 Cubic No

8

M3B7O13X O 4 |T| 3 Cubic No M...X=Cr..Br,

Cr..Cl, Cr..I,

Mn..Br, Ni..I,Co..I

Cu..Br, Cu..I,

Fe..I, β-Mg..Cl

Table S3 Anhydrous borates with the Normal-ring FBB

Chemical

Formula

FBB R Crystal

System

Centro

symmetric

Note

GdBO3 <3T> 0 Trigonal No

β-BaB2O4 <3∆> 0 Trigonal No

YBO3 <3T> 0 Orthorhombic Yes

α-BaB2O4 <3∆> 0 Trigonal Yes

Sr3B3O3N3 <3∆> 0 Monoclinic Yes N is treated as O

MM’B6O12 <3∆> 0 Trigonal Yes MM’=Ba2Ca,

Ba2Co,

Sr1.16Ba1.84,

Ba2Mn

Ba2ZnB6O12 <3∆> 0 Triclinic Yes

AgBO2 <∆2T> + ∆ 1 Orthorhombic Yes

LiBa2B5O10 ∆ + <∆2T> + ∆ 1 Monoclinic Yes

Cu(RE)2B8O16 |∆|∆| +<2∆2T> +|∆|∆| 1 Monoclinic Yes RE=Tb, Dy, Ho

Bi2B8O15 ∆T + <3∆> + ∆T∆ 2 Monoclinic No

La0.785Th0.215

Al2B4O10O0.67

<6T> 2 Hexagonal No

REAl2.03B4O10O0.54 <6T> 2 Hexagonal No RE=La, Nd

*Ba2Al3Nd2B12O27.5 <6T> & <6∆> 2 Hexagonal No

CaAlB3O7 <3T> 2 Orthorhombic Yes

MZnBP2O8 T’ + <2T2T’> + T’ 2 Monoclinic Yes M=K, NH4

Pb3BP3O12 T + <∆2T> + T 2 Orthorhombic Yes

NiHo2B4O10 <4T> 2 Monoclinic Yes

Cu(Ln)2B4O10 <4T> 2 Monoclinic Yes Ln=Tm, Ho, Lu, Er

(RE)4B6O15 2T + TT + 2T 2 Monoclinic Yes RE=Dy, Ho

*Ba2KZn3B9O19 <3∆> & 2<2∆T> 2 Triclinic Yes

M3B3O6 <3∆> 2 Trigonal Yes M=Na, K, Rb, Cs

γ-LiBO2 <6T> 3 Tetragonal No

MB3O5 <2∆T> 3 Orthorhombic No M=Li, Cs, Rb, Tl

CsLiB6O10 <2∆T> 3 Tetragonal No

MM’B9O15 <2∆T> 3 Trigonal No MM’=BaLi,

BaNa, SrLi

α-CsB9O14 <3∆> + <2∆T> +

<3∆>

3 Tetragonal No

9

CsAl4Be4B12O18 <4T> 3 Cubic No

MB2O4 <3T> 3 Tetragonal No M=Pb, Cu

M4B6O13 <4T> 3 Cubic No M=Zn, Co

Li4NaKAl4Be3

B10O27

<3T> 3 Cubic No

Zn6Ga2P6B12O24 <4T> 3 Cubic No

Zn8Se2B12O24 <4T> 3 Cubic No

SrB2O4 <3T> 3 Cubic Yes

CaB2O4(IV) <3T> 3 Cubic Yes

LnBa3B9O18 <3∆> 3 Hexagonal Yes Ln=Pr→Yb

M3B7O12 <2∆T> + <∆2T> + ∆ 3 Triclinic Yes M=Na, Li

β-REB5O9 <∆2T> + |∆|∆| 3 Monoclinic Yes RE=Ce, La

Table S4 Anhydrous borates with the Bridge-ring FBB

Chemical

Formula

FBB R Crystal

System

Centro-

symmetric

Note

Na3GaB4O9 <2∆2T>b 0 Monoclinic Yes

Pb6B10O21 <2∆2T>b + 2∆ +

<2∆2T>b

0 Triclinic Yes

Na5B2P3O13 <2T2T’> + T’ 1 Monoclinic No

β-Dy2B4O9 ∆ + <6T>2b + ∆ 1 Triclinic Yes

Na3B6O9VO4 <3∆3T>B 2 Orthorhombic No

α-(Ln)2B4O9 |T|T| +<14T+TT>B, 2b

+ |T|T|

2 Monoclinic Yes Ln=Eu, Gd, Tb, Dy

Ca2Al2B6O14 <8T>b 2 Orthorhombic Yes

Li2B4O7 <2∆2T>b 3 Tetragonal No

MB4O7 <8T>2b 3 Orthorhombic No M=β-Sr, Pb, Eu, β-Ca

M3B7O13X <6T>B + ∆ ３ Orthorhombic No M=Mn, Fe, Mg, Co,

Ni, X=Cl, Br, I

M3B7O13Cl <6T>B + ∆ 3 Trigonal No M=Mg, Fe,

Li5B7O12.5Cl <6T>B + ∆ 3 Cubic No

LnB3O6 <6T>B 3 Orthorhombic Yes Ln=Dy, Ho, Er, Tm,

Yb, Lu

MB4O7 <2∆2T>b 3 Orthorhombic Yes M=Co, Cd, Mn, Zn

Hg, Mg

β-ZnB4O7 <8T>2b 3 Orthorhombic Yes

α-LnB5O9 <2∆2T>b + ∆ 3 Orthorhombic Yes Ln=Pr, Nd, Sm, Pm, Eu

M3B7O13Cl <6T>B + ∆ 3 Trigonal Yes M=Fe2.4Mg0.6,

Mg1.55Fe1.43Mn0.02

10

Table S5 Anhydrous borates with the ‘8’-shaped-ring FBB

Chemical

Formula

FBB R Crystal

System

Centro

symmetric

Note

CaNa3B5O10 <4∆T>8 0 Triclinic Yes

Li2AlB5O10 <4∆T>8 0 Monoclinic Yes

Bi2B5O12 <3∆2T>8 0 Orthorhombic Yes

BaBPO5 <∆4T’>8 1 Trigonal No

MBPO5 T’ + <2T3T’>n-8 +T’ 1 Trigonal No M= Ca, Pb, Sr

TlBO2 <2∆3T>n-8 + ∆ 1 Tetragonal No

AgBO2 <2∆3T>n-8 + ∆ 1 Orthorhombic Yes

CaB2O4(II) <2∆3T>n-8 + ∆ 1 Orthorhombic Yes

MM’BP2O8 T + <7T>n-8 + T 1 Triclinic Yes MM’=CsZn, RbZn,

(NH4)(Zn0.88Co0.12),

(NH4)Zn

REB3O6 <4∆3T>n-8 1 Orthorhombic Yes RE=La, Pr, Tb, Nd

Li5.5Fe0.5FePb

B12O24

<2∆3T>n-8 + ∆ 1 Trigonal Yes

La2CaB10O19 <2∆3T>8 2 Monoclinic No

MM’B5O10 <2∆3T>8 2 Monoclinic Yes MM’=YCo, LaCo,

YbCo, HoCo,

SmCo,NdCo,

CdLa,CdSm,

CdEu, NiNd,

NiGd, CeZn,

NdZn, TbZn,

LaMg

α-CsB5O8 <4∆T>8 2 Monoclinic Yes

MM’B5O9 <3∆2T>8 2 Monoclinic Yes MM’=NaCa, SrK

Cs3B7O12 <4∆3T>2-8 +

10<3∆2T>8 +

<2∆3T>8 + T

2 Monoclinic Yes

(MM’)2B10O17 <3∆2T>8 2 Monoclinic Yes MM’=CsNa, CsK,

TlNa

CaB2O4(III) <∆4T>8 + ∆ 3 Orthorhombic No

M2B5O9X <2∆3T>8 3 Orthorhombic No M=Ca, Ba, Pb, Eu

X=Cl, Br

Ag2CsB15O24 <4∆T>8 3 Orthorhombic No

β-Tl2B4O7 <2∆3T>8 +

<4∆3T>2-8

3 Triclinic Yes

MB5O8 <4∆T>8 3 Orthorhombic Yes M=α-K, β-K,

α-Rb, β-Rb, Tl,

β-Cs, γ-Cs

Ca2B6O11 <2∆3T>8 + T 3 Monoclinic Yes

11

Table S6 Anhydrous borates with the Combined-rings FBB

Chemical

Formula

FBB R Crystal

System

Centro

symmetric

Note

Rb3B7O12 <3∆2T>8 + T + <2∆T> +

<3∆2T>8

2 Triclinic Yes

AgSrB7O12 <2∆T> + T∆ + <2∆2T>b +

∆T + <2∆T>

2 Monoclinic Yes

β-Na3B9O15 <4∆T>8 + <2∆T> + T 2 Monoclinic Yes

BaB8O13 <4∆T>8 + <∆2T> 3 Tetragonal No

M2M’2B8O14 <4∆T>8 + <∆2T> 3 Orthorhombic No MM’=KLi,

RbLi

BaB4O7 <3∆2T>8 + <∆2T> 3 Monoclinic Yes

Na4B8O14 <3∆2T>8 + <2∆T> 3 Triclinic Yes

Na3B9O15 <4∆T>8 + <2∆T> + T 3 Monoclinic Yes

M2B8O13 <4∆T>8 + <2∆T> 3 Monoclinic Yes M=α-Na,

β-Na, α-Ag,

β-Ag

K3AlB8O15 <4∆T>8 + <2∆T> 3 Monoclinic Yes

Cs3B13O21 2<4∆T>8 + 4(<2∆T> + ∆) 3 Monoclinic Yes

M5B19O31 2(<4∆T>8 + ∆ + <2∆T>)

+T

3 Monoclinic Yes M=K, Rb

M2B4O7 <2∆2T>b + <∆2T> + ∆ 3 Triclinic Yes M=K, Rb

CaB4O7 <2∆2T>b + T + <2∆T> 3 Monoclinic Yes

α-NaB3O5 <4∆T>8 + <2∆2T>b 3 Monoclinic Yes

Na6B13O22.5 <3∆2T>8 + <2∆2T>b +

<2∆2T>b

3 Orthorhombic Yes

Table S1’ Hydrated borates with the Single BO3 or BO4 FBB

Chemical Formula FBB R Crystal

System

Centro

symmetric

Note

Be2BO3(OH)(H2O) ∆ 0 Trigonal No

β-Be2BO3(OH)(H2O) ∆ 0 Hexagonal No

Zn2BO3(OH)0.75F0.25 ∆ 0 Monoclinic No

Ca3Mg(BO3)(CO3)(H2O)x ∆ 0 Cubic No x=0.125, 0.36

Mg6.05Al0.3Fe0.15(BO3)3(OH)4Cl0.4 ∆ 0 Orthorhombic No

Pb5B3O8(OH)3(H2O) ∆ 0 Hexagonal No

NaBF3(OH) ∆ 0 Hexagonal No F is treated as O

Na2BO2(OH) ∆ 0 Orthorhombic Yes

SrBO2(OH) ∆ 0 Orthorhombic Yes

Be2BO3(OH) ∆ 0 Orthorhombic Yes OH=

(OH)0.94F0.06,

12

(OH)0.93F0.07,

(OH)0.7F0.3,

(OH)0.59F0.41,

(OH)0.52F0.48

Cu2BO(OH)5 ∆ 0 Orthorhombic Yes

CdBO3(OH) ∆ 0 Cubic Yes

Mg3BO3(OH)3 ∆ 0 Hexagonal Yes

Mg5O(BO3)(OH)5(H2O)1.4 ∆ 0 Monoclinic Yes

Sr3Ga3O(BO3)4(OH) ∆ 0 Monoclinic Yes

LiNdBO3(OH) T 0 Monoclinic No

LiB(OH)4(H2O)6 T 0 Trigonal No

CsB(OH)4(H2O)2 T 0 Tetragonal No

Ca2Cu(OH)4B2(OH)8 T 0 Triclinic No

SrB2(OH)8 T 0 Triclinic No

Ca2AsO4B(OH)4 T 0 Tetragonal No

MB(OH)4 T 0 Orthorhombic Yes M=Li, Na

Na2B(OH)4Cl T 0 Tetragonal Yes

CuB(OH)4Cl T 0 Tetragonal Yes

NaB(OH)4(H2O)2 T 0 Triclinic Yes

NaHoBO2(OH)3 T 0 Monoclinic Yes

YB(OH)4(CO3) T 0 Orthorhombic Yes

Mn3B(OH)4(PO4)(OH)2 T 0 Orthorhombic Yes

AgBF4(H2O) T 0 Orthorhombic Yes F is treated as O

LiBF4(H2O) T 0 Orthorhombic Yes F is treated as O

Hg2BF4(OH) T 0 Orthorhombic Yes F is treated as O

CaB2(OH)8 T 0 Triclinic Yes

β-CaB2(OH)8 T 0 Orthorhombic Yes

MB2(OH)8(H2O)2 T 0 Monoclinic Yes M=Ca, x=2;

M=Ba, x=1

CaB2O4(H2O)6 T 0 Monoclinic Yes

MB2(OH)8 T 0 Monoclinic Yes M=β-Sr, Ba

BaB2O4(H2O)5 T 0 Monoclinic Yes

Mg3B2(OH)8(SO4)(OH)2 T 0 Orthorhombic Yes (OH)2=(OH)F

MgCa2B2(OH)8(CO3)(H2O)4 T 0 Monoclinic Yes

MgB2(PO4)2(OH)6(H2O)6 T 0 Triclinic Yes

Mg2Al3O(BO4)2(OH) T 0 Monoclinic Yes

CaB2O2(OH)4 T 1 Monoclinic Yes

Table S2’ Hydrated borates with the Branched FBB

Chemical Formula FBB R Crystal

System

Centro

symmetric

Note

K3B3O4(OH)4(H2O)2 2∆T 0 Orthorhombic No

Mg2B2O(OH)6 2T 0 Tetragonal No

13

MgBP2O7(OH)3 T’TT’ 0 Triclinic No

M2B2O4(OH)2 2∆ 0 Monoclinic Yes M2=Mg2,

Mn1.812Mg0.188

Mg2B2O4(H2O)3 2T 0 Tetragonal Yes

CaB2O(OH)6(H2O)2 2T 0 Triclinic Yes

NH4AlBP2O8(OH) T’TT’ 0 Monoclinic Yes

(NH4)5V3BP3O19(H2O) T 3|T’| 0 Triclinic Yes

CsV3B2P4(OH)4O16(H2O)2 T’TT’ 0 Monoclinic Yes

Mn14Mg0.5(B2O5)4(Si0.5O2.2)

(OH)9.8Cl0.8

2∆ 0 Tetragonal Yes

(NH4)6V12B6P12O60(OH)12(H2O)12 T’TT’ 0 Trigonal Yes

MM’BP2O7(OH)3 T’TT’ 1 Monoclinic Yes MM’=NaFe,

NaAl, NaV,

NaGa, NaIn,

KGa, KAl

FeBP2O7(OH)5 T’T 1 Monoclinic Yes

RbMM’BP2O8(OH) T’TT’ 1 Monoclinic Yes MM’=RbV, RbAl,

RbGa, RbFe,

CsFe, CsAl, Pb2,

CsGa, NH4Fe,

NH4Ga, NH4V,

NH4Fe0.47V0.50

Na2BP2O7(OH) T’TT’ 2 Monoclinic No

Table S3’ Hydrated borates with the Normal-ring FBB

Chemical Formula FBB R Crystal

System

Centro

symmetric

Note

KBO2(H2O) <∆2T> 0 Orthorhombic No

M3B3O4(OH)4(H2O)2 <∆2T> 0 Orthorhombic No M=K, x=2;

M=Rb, x=1

Na1.89Ag0.11BP2O7(OH) <3T3T’> 0 Orthorhombic No

Li2B2O4(OH)4 <2T> 0 Monoclinic Yes [3]

Na2B2O4(OH)4(H2O)6 <2T> 0 Triclinic Yes [3]

CaB2O2(OH)4 <3T> + T 0 Monoclinic Yes

MgBPO4(OH)2(H2O)2 <3T’3T> 0 Trigonal Yes

KB3O5(H2O)3 <2∆T> 0 Monoclinic Yes

MB3O3(OH)5(H2O)5 <∆2T> 0 Monoclinic Yes M=Mg, x=5;

M=Ca, x=4

Ca2B3O4(OH)4Cl <∆2T> 0 Monoclinic Yes

MB3O3(OH)5(H2O)5 <∆2T> 0 Triclinic Yes M=β-Mg, x=5

M=Ca, x=1, 2

Ca2B3O3(OH)7(H2O)3 <3T> 0 Triclinic Yes

ZnB3O5(OH)5(H2O) <∆2T> 0 Orthorhombic Yes

14

NaInBP2O8(OH) T’ + <2T2T’> + T’ 0 Monoclinic Yes

MM’BP2O8(OH) T’ + <2T2T’> + T’ 0 Triclinic Yes MM’=RbIn, KIn,

NH4In, KFe

MgB4O4(OH)6(H2O)6 <∆2T> + T 0 Monoclinic Yes

Ca4MgB4O6(OH)6(CO3)2 <4T> 0 Monoclinic Yes

Ca2B4O4(OH)7Cl(H2O)7 <3T> + T 0 Monoclinic Yes

Na6B6O12(H2O)2 <2∆T> 0 Orthorhombic Yes

Na2B6O6(OH)8 <2∆T> 0 Monoclinic Yes

M2B6O11(H2O)15 <∆2T> 0 Triclinic Yes M=Mg, x=15;

M=Ca, x=7, 9

β-Mg2B6O11(H2O)15 <∆2T> 0 Monoclinic Yes

CaMgB6O6(OH)10(H2O)6 <∆2T> 0 Monoclinic Yes

Ca3B6O6(OH)12(H2O)2 <3T> 0 Monoclinic Yes

Ni1.5Mg1.5B3P3O12(OH)6

(H2O)6

<3T’3T> 0 Trigonal Yes

K3PtB7O11(OH)6(H2O)3 |∆|∆| +<∆2T> +|∆|∆| 0 Orthorhombic Yes

BiB4O6(OH)3 <2∆T> + T 1 Triclinic No

CaB3O4(OH)3(H2O) <∆2T> 1 Monoclinic Yes

TlB3O4(OH)2(H2O)0.5 <2∆T> 1 Orthorhombic Yes

Na2B4O7(H2O)10 <4∆> 1 Monoclinic Yes

LiB3PO6(OH)3 <2∆T> + T’ 1 Orthorhombic Yes

(NH4)2B3PO7(OH)2 <2∆T> + T’ 1 Monoclinic Yes

*Ca2SiB5O9(OH)5 <∆2T> & 2T 1 Monoclinic Yes SiO4 is ignored

KB6PO10(OH)4 <∆2T>+T’+<∆2T> 1 Tetragonal Yes

CaMgB6O8(OH)6(H2O)3 <∆2T> 1 Monoclinic Yes

CaB3O5(OH) <∆2T> 2 Orthorhombic No

REB4O6(OH)2Cl <∆2T> + ∆ 2 Monoclinic No RE=Pr, Nd

Ca3B5O6(OH)7Cl(H2O)8 <6∆6T> 2 Monoclinic No

CaB3O4(OH)3 <∆2T> 2 Monoclinic Yes

Co(C2H10N2)B2P3O12(OH) T’ + <T’2T> + T’ 2 Orthorhombic Yes

K3CuB2P4O14(OH)3 T’ + <2TT’> + |T’|T’| 2 Monoclinic Yes

Rb2Co3B4P6O24(OH)2

(H2O)2

T’ + <2TT’> + T’ 2 Orthorhombic Yes

(Cs0.52K0.42)BeAlB11O27(H2O) <4T> 3 Cubic No

Ca16MgB13O17(OH)12)4

Cl6(H2O)28

|T|T| ∆ +<∆2T> +

∆T |∆|∆| + ∆ |T|T| 3 Orthorhombic No

MB2P2O8(OH) T’ + <2TT’> 3 Monoclinic Yes M=Rb, Cs

LnB6O9(OH)3 <3∆3T> 3 Trigonal Yes Ln=Sm→Lu

Ca2B8O13(OH)2 ∆+<2∆T>+T+<∆2T> 3 Triclinic Yes

MgB12O19(H2O)5 <2∆T> + <∆2T> 3 Monoclinic Yes

15

Table S4’ Hydrated borates with the Bridge-ring FBB

Chemical Formula FBB R Crystal

System

Centro

symmetric

Note

(NH4)2B4O5(OH)4(H2O)2 <2∆2T>b 0 Monoclinic No

K2B4O5(OH)4(H2O)2 <2∆2T>b 0 Orthorhombic No

Na2B4O5(OH)4(H2O)x <2∆2T>b 0 Trigonal No x=3, 2.668

K1.67Na0.33B4O5(OH)4(H2O)3 <2∆2T>b 0 Hexagonal No

Ca(RE)2(B4O5(OH)4)2(H2O)8 <2∆2T>b 0 Orthorhombic No MM2’=CaK2,

Ca(NH4)2,

SrRb2, x=8;

MM2’=SrK2,

x=10

β-NiB6O7(OH)6(H2O)5 <3∆3T>B 0 Triclinic No

*K7MnB13O21(OH)6(H2O) <6∆6T>6b & ∆ 0 Orthorhombic No

M2B4O5(OH)4(H2O)x <2∆2T>b 0 Monoclinic Yes M2=NaRb, x=4;

M2=Na2, x=8;

M2=Cs2, x=3

Ca2MnB4O7(OH)6 <2∆2T>b 0 Orthorhombic Yes

Rb2B4O5(OH)4(H2O)3.6 <2∆2T>b 0 Orthorhombic Yes

MB4O5(OH)4(H2O)7 <2∆2T>b 0 Triclinic Yes M=Mn, Mg

MgB6O7(OH)6(H2O)2 <3∆3T>B 0 Orthorhombic Yes

MB6O7(OH)6(H2O)x <3∆3T>B 0 Monoclinic Yes M=Mg, x=3;

M=Mg, x=4;

M=Ni, x=5;

M=Ni(C2H5OH)

x=3.42

Ca4MgAs2(B6O7(OH)6)2

(H2O)14

<3∆3T>B 0 Monoclinic Yes

Na6Mg(B6O7(OH)6)4(H2O)10 <3∆3T>B 0 Monoclinic Yes

M2M’(B6O7(OH)6)2(H2O)4 <3∆3T>B 0 Triclinic Yes M2M’=K2Co,

K2NH4, K2Mg

x=4;

M2M’=Na2Co,

x=8.67

Mg2(B6O7(OH)6)2(H2O)9 <3∆3T>B 0 Trigonal Yes

Na8B12O20(OH)4 <6∆6T>6b 0 Monoclinic Yes

Ag6B12O18(OH)6(H2O)3 <6∆6T>6b 0 Monoclinic Yes

ZnB2O4(H2O)1.12 <6∆6T>6b 0 Trigonal Yes

*Na4Cu3B2P4O15(OH)2

(HPO4)2

T’ + <2T2T’>b

+ T’ & T’

0 Monoclinic Yes

K6UO2B16O24(OH)8(H2O)12 <8∆8T>8b 0 Monoclinic Yes

16

Rb4V6O6B20O32(OH)12

(H2O)0.5

<3∆7T>3B 0 Monoclinic Yes

Na2MgB12O16(OH)8(H2O)4 <3∆3T>B 1 Monoclinic Yes

MB8O13(H2O)2 <3∆3T>B + 2∆ 2 Monoclinic No M=Ca, Sr, Pb

CaB3O5(OH) <2∆4T>2b 2 Monoclinic Yes

Na2B4O6(OH)2 <2∆2T>b 2 Monoclinic Yes

LnB8O11(OH)5 ∆T + <3∆3T>B 2 Monoclinic Yes Ln=La→Nd

SrCaB14O24(OH)6(H2O)5 <3∆3T>B +

<3∆3T>B + 2∆

2 Monoclinic Yes

Co3B7O13F(OH) <3∆3T>B + ∆ 3 Trigonal No

Cd3B7O13(OH) <3∆3T>B + ∆ 3 Trigonal No

*Pb3B10O16(OH)4 <3∆6T>B,3b & ∆ 3 Trigonal No

Table S5’ Hydrated borates with the ‘8’-shaped-ring FBB

Chemical Formula FBB R Crystal

System

Centro

symmetric

Note

KB5O8(H2O)4 <4∆T>8 0 Orthorhombic No

MB5O6(OH)4(H2O)x <4∆T>8 0 Orthorhombic No M=K, Rb, NH4,

x=2;

M=K, x=4

NH4B5O6(OH)4(H2O)2 <4∆T>8 0 Monoclinic No

MB5O6(OH)4(H2O)x <4∆T>8 0 Monoclinic Yes M=Na, Li, x=3;

M=Na, x=0;

M=Cs, β-Cs, Tl,

x=2

NH4B5O8(H2O)4 <4∆T>8 0 Monoclinic Yes

NaCaB5O6(OH)4(H2O)5 <2∆3T>8 0 Triclinic Yes

(NH4)3B15O20(OH)8(H2O)4 3<4∆T>8 0 Monoclinic Yes

(C(NH2)3)3B9O12(OH)6 <6∆3T>2-8 0 Monoclinic Yes

NaBa3Si2B7O16(OH)4 T + <2∆3T>8 + T 0 Monoclinic Yes SiO4 is ignored

TlB2O3(OH)(H2O) ∆ + <2∆3T>n-8 1 Orthorhombic No

MM’BP2O8(H2O)x T’ + <3T4T’>8

+ T’

1 Hexagonal No MM’=NaMn, Fe,

NaMg, NaLi,

NaFe, NaZn, LiCu,

LiCd, x=3;

MM’=NaCd,

x=2.8;

MM’=NaZn, x=1

MM’=KFe, x=2.5

MM’=H0.5Mn1.25,

H0.5Co1.25,

(NH4)0.5Co1.25,

17

x=2.5;

MM’=(NH4)0.4

Fe1.25, x=2.5;

MM’=In, x=0.8

Bi3B6O13(OH) <∆5T>n-8 1 Triclinic No

*Pb6B11O18(OH)9 ∆ + <∆6T>n-8 &

∆2T

1 Trigonal No

*Sr(Ca0.8Sr0.2)(B10O16(O

H)2(B(OH)3)(H2O)

∆ + <4∆T>8 &

<4∆T>8

1 Monoclinic No

LiB2O3(OH)(H2O) ∆ + <3∆4T>n-8 1 Orthorhombic Yes

TlB2O3(OH)(H2O)0.5 ∆ + <2∆3T>n-8 1 Triclinic Yes

Na2B4O6(OH)2(H2O)3 ∆ + <3∆4T>n-8 1 Monoclinic Yes

LiB5O7(OH)2 <4∆T>8 1 Monoclinic Yes

Na2B5O7(OH)3(H2O)2 <3∆2T>8 1 Triclinic Yes

CaNaB5O7(OH)4(H2O)3 <2∆3T>8 1 Monoclinic Yes

Ca2B5O7(OH)5(H2O) <4∆T>8 1 Monoclinic Yes

NaCaB5O9(H2O)5 <2∆3T>8 1 Monoclinic Yes

K3PB5O8(OH)3 T’ + <2∆5T’>n-8 1 Monoclinic Yes

Ba3B6O9(OH)6 T + <5T>n-8 1 Monoclinic Yes

Mg6B11O15(OH)9 T+<4∆5T>3-8 +T 1 Orthorhombic Yes

M2B5O8(OH)(H2O)2 <3∆2T>8 2 Orthorhombic No M2=K2, Na2

*M2B11O16(OH)5(H2O) 2 <3∆2T>8 & ∆ 2 Monoclinic No M2=β-Sr2, Sr2

Sr(Ca0.8Sr0.2)

*KCa4B22O32(OH)10Cl

(H2O)4

<3∆2T>8 & ∆ 2 Triclinic No

NH4B5O7(OH)2(H2O) <4∆T>8 2 Monoclinic Yes

BaB5O8(OH)(H2O) <3∆2T>8 2 Triclinic Yes

MB5O8(OH)(H2O) <3∆2T>8 2 Monoclinic Yes M=Ca, Na2

CeB5O8(OH)NO3(H2O)3 <3∆2T>8 2 Monoclinic Yes

NaCaB5O8(OH)2(H2O)3 <2∆3T>8 2 Monoclinic Yes

Na3B5O8(OH)2(H2O) <2∆3T>8 2 Monoclinic Yes

Ba2B5O8(OH)3 <2∆3T>8 2 Monoclinic Yes

Na2Ca3B5O8(OH)2(SO4)2Cl <2∆3T>8 2 Orthorhombic Yes

LiBa2B10O16(OH)3 <3∆2T>8 + <2∆3T>8 2 Triclinic Yes

Ba5B20O33(OH)4(H2O) <2∆3T>8 2 Monoclinic Yes

Li3B5O8(OH)2 <2∆3T>8 3 Tetragonal No

Na3B5O9(H2O) <2∆3T>8 3 Orthorhombic No

MM’B5O9Cl <2∆3T>8 3 Triclinic No MM’=Ca2, CaSr

β-Ca2B5O9Cl(H2O) <2∆3T>8 3 Monoclinic No

Sr2B5O9(OH)(H2O) <2∆3T>8 3 Monoclinic No

Ba2B5O9Cl(H2O)0.5 <2∆3T>8 3 Orthorhombic No

Na2Ba2B10O17(OH)2 <2∆3T>8 3 Monoclinic No

Ca2MB9O13(OH)6Cl

(H2O)4

<2∆3T>8 +

|∆T|∆T| 3 Monoclinic No M=Fe,

Fe0.67Mg0.33

18

Ca9B26O34(OH)24Cl4

(H2O)13

∆ + <7∆8T>8 3 Triclinic No

β-Ca9B26O34(OH)24Cl4

(H2O)13

∆ + <7∆8T>8 3 Monoclinic No

Na0.5Pb2B5O9(OH)0.5Cl <4∆T>8 3 Orthorhombic Yes

Pb2B5O9(OH)(H2O)x <2∆3T>8 3 Monoclinic Yes x=0.5, 1

HKMg2B12O16(OH)10(H2O)4 ∆ + <2∆3T>8 3 Monoclinic Yes

Table S6’. Hydrated borates with the ‘8’-shaped-ring FBB

Chemical Formula FBB R Crystal

System

Centro

symmetric

Note

Tl4Cu3B18O28(OH)8

(H2O)10

<2(<2∆2T>b + ∆ +

<2∆2T>b)>

0 Orthorhombic No

K5HCu4B20O32(OH)8

(H2O)33

<4(<2∆2T>b + ∆)> 0 Triclinic Yes

Na6Cu2B16O24(OH)10

(H2O)12

<2(<3∆4T>2-8 +∆)> 0 Triclinic Yes

LnB9O13(OH)4(H2O) <3∆3T>B + <∆2T> 2 Monoclinic Yes Ln=Pr→Eu

NaCa2B9O14(OH)4

(H2O)2

<2∆2T>b +

<2∆3T>8

2 Monoclinic Yes

Na10HB9O14(OH)4

(H2O)2

<2∆2T>b +

<2∆3T>8

2 Triclinic Yes

Footnote:

[1] R may take the value 0, 1, 2 or 3 if the FBB forms isolated, chain, sheet or three-dimensional

network.

[2] Note gives the following information: the compounds having the same structure and the other

treatments about the compound.

[3] In the structures of Na2B2O4(OH)4(H2O)6 (ICSD 27488) and Li2B2O4(OH)4 (ICSD 100854),

two BO4 tetrahedra are connected into a ring by two O-O bonds (O22−), not by sharing an edge.

Therefore, the FBBs of them are treated as normal-rings in the current topological classification.

The corresponding configuration of this FBB is given in Figure S4f.

19

Figures S2 to S7 give some configurations of FBBs and their corresponding topology figures. The principles of topology figure follow the reference by Burns et al. (1995).

Figure S2 Structure and topology of BO3 and BO4 (all the H atoms are omitted for clarity).

Figure S3 Structure and topology of branched FBB (all the H atoms are omitted for clarity).

20

(a)

(b)

(c)

(d)

(e)

(f)

Figure S4

Structure and topology of normal-ring FBBs (all the H atoms are omitted for clarity):

(a) 3-normal-ring FBBs; (b) 4-normal-ring FBBs; (c) 6-normal-ring FBB; (d) tailed

normal-ring FBBs; (e) the linear combination of normal-ring FBB; (f) the

configuration and topology of FBB of Na2B2O4(OH)4(H2O)6 (ICSD 27488) and

Li2B2O4(OH)4 (ICSD 100854).

21

(a)

(b)

(c)

(d)

(e) Figure S5 Structure and topology of bridge-ring FBBs (all the H atoms are omitted for clarity): (a) typical bridge-ring FBB; (b) the linear combination of bridge-ring FBB; (c) double-bridge-ring FBB; (d) 6-bridge-ring FBB; (e) big-bridge-ring FBBs.

22

(a)

(b)

(c)

Figure S6

Structure and topology of ‘8’-shaped-ring FBB (all the H atoms are omitted for

clarity): (a) typical ‘8’-shaped-ring FBBs; (b) double-’8’-shaped-ring FBB; (c) the

linear combination of ‘8’-shaped-ring FBBs.

23

(a)

(b)

Figure S7 Structure and topology of combined-ring FBBs (all the H atoms are omitted for clarity): (a) the linear combination of combined-ring FBBs; (b) the mosaic combination of combined-ring FBBs.

24

Table S7 The number of anhydrous borates and the number of non-centrosymmetric anhydrous (behind the /) borates in each kind

Single ∆ or

T Branched Normal

-ring Bridge-ring

‘8’-shaped-ring

Combined-ring

Total

Isolate 264/68 49/12 10/2 2/0 3/0 0 328/82

Chain 5/1 6/4 5/0 2/1 16/5 0 34/11

Layer Non-existed 2/1 20/5 6/1 23/1 3/0 54/8

Network Non-existed 13/13 37/19 43/23 17/8 19/3 129/66

Total 269/69 70/30 72/26 53/25 59/14 22/3 545/167

Table S8 The number of hydrated borates and the number of non-centrosymmetric hydrated (behind the /) borates in each kind

Single ∆ or

T Branched Normal

-ring Bridge-ring

‘8’-shaped-ring

Combined-ring

Total

Isolate 51/14 12/3 35/4 36/11 17/6 3/1 154/39

Chain 1/0 21/1 9/1 1/0 32/21 0 64/23

Layer Non-existed 1/1 9/4 10/3 16/6 7/0 43/14

Network Non-existed 0 16/2 3/3 16/12 0 35/17

Total 52/14 34/5 69/11 50/17 81/45 10/1 296/93

FBB

Linkage

Linkage

FBB

25

Figure S7 Classes of the 6 topological structures of FBBs (by our classification) involved in 841 borates; the percentages are calculated for the occurrence of each kind of borate. Figure S8 Classes of the 4 degrees of polymerization of FBBs (by other classifications) involved in 841 borates; the percentages are calculated for the occurrence of each kind of borate.

Single ∆ or T borates

(36.7%)

Combined-ring

(3.3%)

‘8’-shaped-rin

g

Branched borates

(12.4%)

Normal-ring

(16.3%)

Bridge-ring

(13.3%)

Layered borates

(10.7%)

Isolate borates

(54.8%)

Chained borates

(11.9%)

Network borates

(22.6%)