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# U+210DDouble-Struck Capital H

U+210D was added in Unicode version 1.1 in 1993. It belongs to the block U+2100 to U+214F Letterlike Symbols in the U+0000 to U+FFFF Basic Multilingual Plane.

This character is a Uppercase Letter and is commonly used, that is, in no specific script.

The glyph is a font version of the glyph Latin Capital Letter H. It has no designated width in East Asian texts. In bidirectional text it is written from left to right. When changing direction it is not mirrored. The word that U+210D forms with similar adjacent characters prevents a line break inside it. The glyph can be confused with one other glyph.

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by $H.$ Although multiplication of quaternions is noncommutative, it gives a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form

where the coefficients a, b, c, d are real numbers, and 1, i, j, k are the basis vectors or basis elements.

Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, magnetic resonance imaging and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.

In modern terms, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, also a division ring and a domain. It is a special case of a Clifford algebra, classified as ${\mathrm{Cl}}_{0,2}\left(R\right)\cong {\mathrm{Cl}}_{3,0}^{+}\left(R\right).$ It was the first noncommutative division algebra to be discovered.

According to the Frobenius theorem, the algebra $H$ is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. The next extension gives the sedenions, which have zero divisors and so cannot be a normed division algebra.

The unit quaternions give a group structure on the 3-sphere S3 isomorphic to the groups Spin(3) and SU(2), i.e. the universal cover group of SO(3). The positive and negative basis vectors form the eight-element quaternion group.

## Representations

System Representation
8461
UTF-8 E2 84 8D
UTF-16 21 0D
UTF-32 00 00 21 0D
URL-Quoted %E2%84%8D
HTML hex reference &#x210D;
Wrong windows-1252 Mojibake â„
HTML named entity &Hopf;
HTML named entity &quaternions;
LATEX \mathbb{H}

## Complete Record

Property Value
Age (age) 1.1 (1993)
Unicode Name (na) DOUBLE-STRUCK CAPITAL H
Unicode 1 Name (na1) DOUBLE-STRUCK H
Block (blk) Letterlike Symbols
General Category (gc) Uppercase Letter
Script (sc) Common
Bidirectional Category (bc) Left To Right
Combining Class (ccc) Not Reordered
Decomposition Type (dt) font
Decomposition Mapping (dm) Latin Capital Letter H
Lowercase (Lower)
Simple Lowercase Mapping (slc) Double-Struck Capital H
Lowercase Mapping (lc) Double-Struck Capital H
Uppercase (Upper)
Simple Uppercase Mapping (suc) Double-Struck Capital H
Uppercase Mapping (uc) Double-Struck Capital H
Simple Titlecase Mapping (stc) Double-Struck Capital H
Titlecase Mapping (tc) Double-Struck Capital H
Case Folding (cf) Double-Struck Capital H
ASCII Hex Digit (AHex)
Alphabetic (Alpha)
Bidi Control (Bidi_C)
Bidi Mirrored (Bidi_M)
Composition Exclusion (CE)
Case Ignorable (CI)
Changes When Casefolded (CWCF)
Changes When Casemapped (CWCM)
Changes When NFKC Casefolded (CWKCF)
Changes When Lowercased (CWL)
Changes When Titlecased (CWT)
Changes When Uppercased (CWU)
Cased (Cased)
Full Composition Exclusion (Comp_Ex)
Default Ignorable Code Point (DI)
Dash (Dash)
Deprecated (Dep)
Diacritic (Dia)
Emoji Modifier Base (EBase)
Emoji Component (EComp)
Emoji Modifier (EMod)
Emoji Presentation (EPres)
Emoji (Emoji)
Extender (Ext)
Extended Pictographic (ExtPict)
FC NFKC Closure (FC_NFKC) Latin Small Letter H
Grapheme Cluster Break (GCB) Any
Grapheme Base (Gr_Base)
Grapheme Extend (Gr_Ext)
Hex Digit (Hex)
Hyphen (Hyphen)
ID Continue (IDC)
ID Start (IDS)
IDS Binary Operator (IDSB)
IDS Trinary Operator and (IDST)
IDSU (IDSU) 0
ID_Compat_Math_Continue (ID_Compat_Math_Continue) 0
ID_Compat_Math_Start (ID_Compat_Math_Start) 0
Ideographic (Ideo)
InCB (InCB) None
Indic Mantra Category (InMC)
Indic Positional Category (InPC) NA
Indic Syllabic Category (InSC) Other
Jamo Short Name (JSN)
Join Control (Join_C)
Logical Order Exception (LOE)
Math (Math)
Noncharacter Code Point (NChar)
NFC Quick Check (NFC_QC) Yes
NFD Quick Check (NFD_QC) Yes
NFKC Casefold (NFKC_CF) Latin Small Letter H
NFKC Quick Check (NFKC_QC) No
NFKC_SCF (NFKC_SCF) Latin Small Letter H
NFKD Quick Check (NFKD_QC) No
Other Alphabetic (OAlpha)
Other Default Ignorable Code Point (ODI)
Other Grapheme Extend (OGr_Ext)
Other ID Continue (OIDC)
Other ID Start (OIDS)
Other Lowercase (OLower)
Other Math (OMath)
Other Uppercase (OUpper)
Prepended Concatenation Mark (PCM)
Pattern Syntax (Pat_Syn)
Pattern White Space (Pat_WS)
Quotation Mark (QMark)
Regional Indicator (RI)
Sentence Break (SB) Upper
Soft Dotted (SD)
Sentence Terminal (STerm)
Terminal Punctuation (Term)
Unified Ideograph (UIdeo)
Variation Selector (VS)
Word Break (WB) Alphabetic Letter
White Space (WSpace)
XID Continue (XIDC)
XID Start (XIDS)
Expands On NFC (XO_NFC)
Expands On NFD (XO_NFD)
Expands On NFKC (XO_NFKC)
Expands On NFKD (XO_NFKD)
Bidi Paired Bracket (bpb) Double-Struck Capital H
Bidi Paired Bracket Type (bpt) None
East Asian Width (ea) neutral
Hangul Syllable Type (hst) Not Applicable
ISO 10646 Comment (isc)
Joining Group (jg) No_Joining_Group
Joining Type (jt) Non Joining
Line Break (lb) Alphabetic
Numeric Type (nt) none
Numeric Value (nv) not a number
Simple Case Folding (scf) Double-Struck Capital H
Script Extension (scx)
Vertical Orientation (vo) R